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Grasshopper is a graphical algorithm editor tightly integrated with Rhino’s 3-D modeling tools, allowing
designers to build form generators from the simple to the awe-inspiring.
You have just opened the third edition of the Grasshopper Primer. This primer was originally written by Andrew O.
Payne of Lift Architects for Rhino4 and Grasshopper version 0.6.0007 which, at the time of its release, was a giant
upgrade to the already robust Grasshopper platform. We now find ourselves at another critical shift in Grasshopper
development, so a much needed update to the existing primer was in order. We are thrilled to add this updated, and
now web-based, primer to the many amazing contributions put forth by Grasshopper community members.
With an already excellent foundation from which to build, our team at Mode Lab went to work designing and
developing the look and feel of the third edition. This revision provides a comprehensive guide to the most current
Grasshopper build, version 0.90076, highlighting what we feel are some of the most exciting feature updates. The
revised text, graphics, and working examples are intended to teach visual programming to the absolute beginner,
as well as provide a quick introduction to Generative Design workflows for the seasoned veteran. It is our goal that
this primer will serve as a field guide to new and existing users looking to navigate the ins and outs of using
Grasshopper in their creative practice.
This primer introduces you to the fundamental concepts and essential skill-building workflows to effectively use
Grasshopper. Foundations is the first volume in a forthcoming collection of Grasshopper primers. Here's what you
can expect to learn from the primer:
Introduction - What is Grasshopper and how is it being used?
Hello Grasshopper - Make your first definition
Anatomy of a Grasshopper Definition - What makes up a definition?
Building Blocks of Algorithms - Start simple and build complexity
Designing with Lists - What's a list and how do I manage them?
Designing with Data Trees - What's a data structure and what do they mean for my process?
The Grasshopper Primer (EN)
Third Edition V3.3
WELCOME
The Grasshopper Primer V3.3
1About
http://modelab.is
Appendix - References and Working files for continued exploration
We hope that at the very least this primer will inspire you to begin exploring the many opportunities of programming
with Grasshopper. We wish you the best of luck as you embark on this journey.
The Grasshopper Primer is an open source project, initiated by Bob McNeel, Scott Davidson, and the Grasshopper
Development team at Robert McNeel & Associates.
Mode Lab authored the Third Edition of the primer. http://modelab.is
If you would like to contribute to this project, check out the github project wiki to get started
(https://github.com/modelab/grasshopper-primer/wiki).
A special thanks to David Rutten for the endless inspiration and invaluable work pioneering Grasshopper. We
would also like to thank Andrew O. Payne for providing the assets from which this work initiated. Lastly, many
thanks to Bob McNeel and everyone at Robert McNeel & Associates for their generous support over the years.
Rhino5
Rhino 5.0 is the market leader in industrial design modeling software. Highly complicated shapes can be directly
modeled or acquired through 3D digitizers. With its powerful NURBS based engine Rhino 5.0 can create, edit,
analyze, and translate curves, surfaces, and solids. There are no limits on complexity, degree, or size.
http://www.rhino3d.com/download/rhino/5/latest
Grasshopper
For designers who are exploring new shapes using generative algorithms, Grasshopper is a graphical algorithm
editor tightly integrated with Rhino’s 3D modeling tools. Unlike RhinoScript or Python, Grasshopper requires no
knowledge of the abstract syntax of scripting, but still allows designers to build form generators from the simple to
the awe inspiring.
http://www.grasshopper3d.com/page/download-1
THE GRASSHOPPER PRIMER PROJECT
ACKNOWLEDGEMENTS
REQUIRED SOFTWARE
The Grasshopper Primer V3.3
2About
http://www.en.na.mcneel.com/
http://modelab.is
https://github.com/modelab/grasshopper-primer/wiki
http://www.rhino3d.com/download/rhino/5/latest
http://www.grasshopper3d.com/page/download-1
The Grasshopper forum is very active and offers a wonderful resource for posting questions/answers and finding
help on just about anything. The forum has categories for general discussion, errors & bugs, samples & examples,
and FAQ.
http://www.grasshopper3d.com/forum
The Common Questions section of the Grasshopper site contains answers to many questions you may have, as
well as helpful links:
http://www.grasshopper3d.com/notes/index/allNotes
For more general questions pertaining to Rhino3D be sure to check out the McNeel Forum powered by Discourse.
http://discourse.mcneel.com/
The Grasshopper Primer is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0
Unported license. The full text of this license is available here: http://creativecommons.org/licenses/by-nc-
sa/3.0/us/legalcode
FORUMS
LICENSING INFORMATION
The Grasshopper Primer V3.3
3About
http://www.grasshopper3d.com/forum
http://www.grasshopper3d.com/notes/index/allNotes
http://discourse.mcneel.com/
http://creativecommons.org/licenses/by-nc-sa/3.0/us/legalcode
0. About
0.1. Grasshopper - an Overview
0.2. Grasshopper in Action
1. Foundations
1.1. Hello Grasshopper!
1.1.1. Installing and Launching Grasshopper
1.1.2. The Grasshopper UI
1.1.3. Talking to Rhino
1.2. Anatomy of a Grasshopper Definition
1.2.1. Grasshopper Object Types
1.2.2. Grasshopper Component Parts
1.2.3. Data Types
1.2.4. Wiring Components
1.2.5. The Grasshopper Definition
1.3. Building Blocks of Algorithms
1.3.1. Points Planes & Vectors
1.3.2. Working With Attractors
1.3.3. Mathematics, Expressions & Conditionals
1.3.4. Domains & Color
1.3.5. Booleans and Logical Operators
1.4. Designing with Lists
1.4.1. Curve Geometry
1.4.2. What is a List?
1.4.3. Data Stream Matching
1.4.4. Creating Lists
1.4.5. List Visualization
1.4.6. List Management
1.4.7. Working with Lists
1.5. Designing with Data Trees
1.5.1. Surface Geometry
1.5.2. What is a Data Tree?
1.5.3. Creating Data Trees
1.5.4. Working with Data Trees
1.6. Getting Started with Meshes
1.6.1. What is a Mesh?
1.6.2. Understanding Topology
1.6.3. Creating Meshes
1.6.4. Mesh Operations
1.6.5. Mesh Interactions
1.6.6. Working with Mesh Geometry
2. Extensions
2.1. Element*
2.1.1. Introduction
2.1.2. Half Edge Data
2.1.3. Components
2.1.4. Exercise
2.1.5. Architectural Case Study
Table of Contents
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3. Appendix
3.1. Index
3.2. Example Files
3.3. Resources
3.4. About This Primer
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5
The origins of Grasshopper can be traced to the functionality of Rhino3d Version 4’s “Record History” button. This
built-in feature enabled users to store modeling procedures implicitly in the background as you go. If you lofted four
curves with the recording on and then edited the control points of one of these curves, the surface geometry would
update. Back in 2008, David posed the question: “what if you could have more explicit control over this history?” and
the precursor to Grasshopper, Explicit History, was born. This exposed the history tree to editing in detail and
empowered the user to develop logical sequences beyond the existing capabilities of Rhino3D’s built in features.
Six years later, Grasshopper is now a robust visual programming editor that can be extended by suites of externally
developed add-ons. Furthermore, it has fundamentally altered the workflows of professionals across multiple
industries and fostered an active global community of users.
This primer focuses on Foundations, offering the core knowledge you need to dive into regular use of Grasshopper
and several on-ramps to how you might go further within your own creative practice. Before diving into the
descriptions, diagrams, and examples suppliedhereafter, let’s discuss what visual programming is, the basics of
the Grasshopper interface and terminology, as well as the “live” characteristics of the viewport feedback and user
experience.
Visual Programming is a paradigm of computer programming within which the user manipulates logic elements
graphically instead of textually. Some of the most well-known textual programming languages such as C#, Visual
Basic, Processing – and more close to home for Rhino – Python and Rhinoscript require us to write code that is
bound by language-specific syntax. In contrast, visual programming allows us to connect functional blocks into a
sequence of actions where the only “syntax” required is that the inputs of the blocks receive the data of the
appropriate type, and ideally, that is organized according to the desired result – see the sections on Data Stream
Matching and Designing with Data Trees. This characteristic of visual programming avoids the barrier to entry
commonly found in trying to learn a new language, even a spoken one, as well as foregrounds the interface, which
for designers locates Grasshopper within more familiar territory.
Grasshopper - an Overview
Grasshopper is a visual programming editor developed by David Rutten at Robert McNeel & Associates. As a
plug-in for Rhino3D, Grasshopper is integrated with the robust and versatile modeling environment used by
creative professionals across a diverse range of fields, including architecture, engineering, product design,
and more. In tandem, Grasshopper and Rhino offer us the opportunity to define precise parametric control over
models, the capability to explore generative design workflows, and a platform to develop higher-level
programming logic – all within an intuitive, graphical interface.
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6Grasshopper - an Overview
This image show the process for drawing a sine curve in python and in Grashopper.
To access Grasshopper and its visual programming capabilities, we need to download and install the program
from the Grasshopper3D.com website. Once installed, we can open the plug-in by typing “Grasshopper” into the
Rhino Command Line. The first time we do so in a new session of Rhino, we will be presented with the
Grasshopper loading prompt followed by the Grasshopper editor window. We can now add functional blocks called
“components” to the “canvas,” connect them with “wires,” and save the entire “definition” in the .ghx file format.
A Grasshopper definition, made up of components connected with wires on the canvas
Once we’ve started to develop a Grasshopper definition and created “slider” objects within our canvas to control
geometry, we may naturally intuit the connection we’ve made between this input object to what we see in Rhino’s
viewports. This connection is essentially live – if we adjust the grip on the slider, we will see the consequences in
that, within our definition an input somewhere has changed and the program must be solved again to recompute a
solution and display the update. To our benefit when getting started with using Grasshopper, the geometry preview
we see is a lightweight representation of the solution and it automatically updates. It is important to take note this
connection now as when your definitions become more complex, adeptly managing the flow of data, the status of
the “solver,” and what is previewed in the Rhino viewport will prevent many unwanted headaches.
Program flow from left to right
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7Grasshopper - an Overview
Grasshopper is a graphical algorithm editor that is integrated with Rhino3D’s modeling tools.
Algorithms are step by step procedures designed to perform an operation.
You use Grasshopper to design algorithms that then automate tasks in Rhino3D.
An easy way to get started if you are unclear how to perform a specific operation in Grasshopper would be to try
manually and incrementally creating an algorithm using Rhino commands.
As you begin first exploring Grasshopper or further building your skills, you have joined the global Grasshopper
community, one that is full of active members from many fields, with diverse experience levels. The forum at
Grasshopper3D. com is a useful resource for posing questions, sharing findings, and gaining knowledge. This is
a community that we have held dear as we’ve written this primer and watched Grasshopper develop over the years.
Welcome!
THINGS TO REMEMBER
The Grasshopper Primer V3.3
8Grasshopper - an Overview
Follow the Grasshopper in Action board on Pinterest.
Grasshopper in Action
The Grasshopper Primer V3.3
9Grasshopper in Action
http://www.pinterest.com/modelabnyc/grasshopper-in-action/
http://www.pinterest.com/modelabnyc/grasshopper-in-action/
1. FOUNDATIONS
A strong foundation is built to last. This volume of the Primer introduces the
key concepts and features of parametric modeling with Grasshopper.
The Grasshopper Primer V3.3
10Foundations
1.1. HELLO GRASSHOPPER
Grasshopper is a graphical algorithm editor that is integrated with Rhino3D’s
modeling tools. You use Grasshopper to design algorithms that then automate
tasks in Rhino3D.
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11Hello Grasshopper!
To download the Grasshopper plug-in, visit the Grasshopper web site. Click on the Download tab at the top of the
page, and when prompted on the next screen, enter your email address. Now, right click on the download link, and
choose Save Target As from the menu. Select a location on your hard drive (note: the file cannot be loaded over a
network connection, so the file must be saved locally to your computer’s hard drive) and save the executable file to
that address.
Download Grasshopper from the grasshopper3d.com website.
Select Run from the download dialog box follow the installer instructions. (note: you must have Rhino 5 already
installed on your computer for the plug-in to install properly).
1.1.1. INSTALLING AND LAUNCHING GRASSHOPPER
The Grasshopper plugin is updated frequently so be sure to update to the latest build.
Note that there is currently no version of Grasshopper for Mac.
1.1.1.1. DOWNLOADING
1.1.1.2. INSTALLING
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12Install ing and Launching Grasshopper
http://grasshopper3d.com
Follow the steps in the Installation wizard.
To Launch Grasshopper, type Grasshopper into the Rhino Command line. When you launch Grasshopper, the first
thing you will see is a new window floating in front of Rhino. Within this window you can create node-based
programs to automate various types of functionality in Rhino. Best practice is to arrange the windows so that they
do not overlap and Grasshopper does not obstruct the Rhino viewports.
Type “Grasshopper” into the Rhino command line to launch the Grasshopper plugin.
1.1.1.3. LAUNCHING
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13Install ing and Launching Grasshopper
1. The Grasshopper window floats on top of the Rhino viewports.
2. Grasshopper displays the version number at the bottom of the window.
Split the screen so that Grasshopper does not obstruct the Rhino Viewports. You can do this by dragging
each window to opposite sides of the screen, or by holding the Wondows key and pressing the left or right
arrows.
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14Install ing and Launching Grasshopper
Let’s start by exploring Grasshopper’s user interface UI. Grasshopper is a visual programming application where
you are able to create programs, called definitions or documents, by dragging components onto the main editing
window (called the canvas). The outputs to these components are connected to the inputs of subsequent
components — creating a graph of information which can be read from left to right. Let’s get started with the basics.
Assuming you’ve already installed the Grasshopper plugin (see F.0.0), type the word “Grasshopper” in the Rhino
command prompt to display the Grasshopper Editor. The Grasshopper interface contains a number of elements,
most of which will be very familiar to Rhino users. Let’s look at a few features ofthe interface.
1. Windows title bar.
2. Main menu bar.
3. File browser control.
4. Component palettes.
5. Canvas toolbar.
6. Canvas.
7. This area, indicated by a grid of rectangular boxes, provides an interface with which to open recently
accessed file. The 3x3 menu shows the files most recently accessed (chronologically) and will display a
red rectangular box if the file cannot be found (which can occur if you move a file to a new folder or delete
it).
1.1.2. THE GRASSHOPPER UI
Grasshopper’s visual “plug-and-play” style gives designers the ability to combine creative problem solving with
novel rule systems through the use of a fluid graphical interface.
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15The Grasshopper UI
8. The status bar tells you what version of Grasshopper is currently installed on your machine. If a newer
version is available, a pop-up menu will appear in your tray providing instructions on how to download
the latest version.
The Editor Window title bar behaves differently from most other dialogs in Microsoft Windows. If the window is not
minimized or maximized, double clicking the title bar will collapse the dialog into a minimized bar on your screen.
This is a great way to switch between the plug-in and Rhino because it minimizes the Editor without moving it to the
bottom of the screen or behind other windows. Note that if you close the Editor, the Grasshopper geometry preview
in the Rhino viewport will disappear, but the file won’t actually be closed. The next time you run the “Grasshopper”
command in the Rhino dialog box, the window will come back in the same state with the same files loaded. This is
because once it is launched from the command prompt, your session of Grasshopper stays active until that
instance of Rhino is closed.
The title bar is similar to typical Windows menus, except for the file browser control on the right (see next section).
The File menu provides typical functions (eg. New File, Open, Save, etc.) in addition to a few utility tools which let
you export images of your current Grasshopper document (see Export Quick Image and Export Hi-Res Image). You
can control different aspects of the user interface using the View and Display menus, while the Solution menu lets
you manage different attributes about how the solver computes the graph solution.
It is worth noting that many application settings can be controlled through the Preferences dialog box found under
the File menu. The Author section allows you to set personal metadata which will be stored with each Grasshopper
document while the Display section gives you fine grain control over the look and feel of the interface. The Files
section lets you specify things like how often and where to store automatically saved file (in case the application is
inadvertently closed or crashes). Finally, the Solver section lets you manage core and third-party plugins which can
extend functionality.
Note: Be careful when using shortcuts since they are handled by the active window which could either be
Rhino, the Grasshopper canvas or any other window inside Rhino. It is quite easy to use a shortcut
command, only to realize that you had the wrong active window selected and accidentally invoked the wrong
command.
1.1.2.1. THE WINDOWS TITLE BAR
1.1.2.2. MAIN MENU BAR
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16The Grasshopper UI
The Preferences dialog allows you to set many of Grasshopper’s application settings.
The File Browser allows you to quickly switch between different loaded files by selecting them through this drop-
down list. Accessing your open files through the File Browser drop-down list enables you to quickly copy and paste
items from open definitions. Just click on the active file name in the browser control window and a cascading list of
all open files will be displayed (along with a small thumbnail picture of each open definition) for easy access. You
can also hit Alt+Tab to quickly switch between any open Grasshopper documents.
Of course, you can go through the standard Open File dialog to load any Grasshopper document, although you can
also drag and drop any Grasshopper file onto the canvas to load a particular definition.
Grasshopper is a plug-in that works “on-top” of Rhino and as such has its own file types. The default file type
is a binary data file, saved with an extension of .gh. The other file type is known as a Grasshopper XML file,
which uses the extension .ghx. The XML (Extensible Markup Language) file type uses tags to define objects
and object attributes (much like an .HTML document) but uses custom tags to define objects and the data
within each object. Because XML files are formatted as textdocuments, you could open up any Grasshopper
XML file in a text editor like NotePad to see the coding that is going on behind the scenes.
Grasshopper has several different methods by which it can open a file, and you will need to specify which option
you would like to use when using this method.
Open File: As the name suggests, this file option will simply open any definition that you drag and drop onto the
canvas.
1.1.2.3. FILE BROWSER CONTROL
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17The Grasshopper UI
Insert File: You can use this option to insert an existing file into the current document as loose components.
Group File: This method will insert a file into an existing document, but will group all of the objects together.
Cluster File: Similar to the group function, this method will insert a file into an existing document, but will create a
cluster object for the entire group of objects.
Examine File: Allows you to open a file in a locked state, meaning you can look around a particular file but you can’t
make any changes to the definition.
Grasshopper also has an Autosave feature which will be triggered periodically based on specific user actions. A list
of Autosave preferences can be found under the File menu on the Main Menu Bar. When the active instance of
Rhino is closed, a pop-up dialog box will appear asking whether or not you want to save any Grasshopper files that
were open when Rhino was shut down.
Autosave only works if the file has already been saved at least once.
Drag and Drop Files onto the Canvas.
This area organizes components into categories and sub-categories. Categories are displayed as tabs, and
subcategories are displayed as drop-down panels. All components belong to a certain category. These categories
have been labeled to help you find the specific component that you are looking for (e.g. “Params” for all primitive
1.1.2.4. COMPONENT PALETTES
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18The Grasshopper UI
data types or “Curves” for all curve related tools). To add a component to the canvas, you can either click on the
objects in the drop-down menu or you can drag the component directly from the menu onto the canvas.
Drag + Drop a component from the palette to add a component to the canvas.
Since there can be many more components in each sub-category than will fit into the palette, a limited number of
icons are displayed on each panel. The height of the component palette and the width of the Grasshopper window
can be adjusted to display more or fewer components per sub-category. To see a menu of all of the components in
a given sub-category, simply click on the black bar at the bottom of each sub-category panel. This will open a
dropdown menu which provides access to all components in that sub-category.
1. Catgory tab
2. Sub-category panel.
3. Click the black bar to open the sub-category panel menu.
4. Hover your mouse over a component for a short description.
5. Drop-down menu.
1.1.2.5. THE CANVAS
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19The Grasshopper UI
The canvas is the primary workspace for creating Grasshopper definitions. It is here where you interact with the
elements of your visual program. You can start working in the canvas by placing components and connecting wires.
Grouping components together on the canvas can be especially useful for readability and comprehensibility.
Groupingallows you the ability to quickly select and move multiple components around the canvas. You can create
a group by typing Ctrl+G with the desired components selected. An alternate method can be found by using the
“Group Selection” button under the Edit Menu on the Main Menu Bar. Custom parameters for group color,
transparency, name, and outline type can be defined by right-clicking on any group object.
1. A group of components delineated by the Box Outline profile.
2. Right-click anywhere on the group to edit the name and appearance of the group.
You can also define a group using a meta-ball algorithm by using the Blob Outline profile.
Two groups are nested inside one another. The color (light blue) has been changed on the outer group to
help visually identify one group from the other. Groups are drawn “behind” the components within them and,
in cases such as this, there is a depth order to the two groups. To change this, go to Edit > Arrange in the
1.1.2.6. GROUPING
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20The Grasshopper UI
main menu bar.
There are a few widgets that are available in Grasshopper that can help you perform useful actions. You can toggle
any of these widgets on/off under the Display menu of the Main Menu bar. Below we’ll look at a few of the most
frequently used widgets.
The Align Widget One useful UI widget which can help you keep your canvas clean is the Align widget. You can
access the Align widget by selecting multiple components at the same time and clicking on one of the options
found in the dashed outline that surrounds your selected components. You can align left, vertical center, right, or top,
horizontal center, bottom, or distribute components equally through this interface. When first starting out, you may
find that these tools sometimes get in the way (it is possible to make the mistake of collapsing several
components on top of each other). However, with a little practice these tools can be invaluable as you begin to
structure graphs which are readable and comprehensible.
1. Align right.
2. Distribute vertically.
The Profiler Widget The profiler lists worst-case runtimes for parameters and components, allowing you to track
down bottlenecks in networks and to compare different components in terms of performance. Note that this widget
is turned off by default.
The Profiler widget gives you visual feedback as to which components in your definition could be causing
longer computational times.
1.1.1.7. WIDGETS
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21The Grasshopper UI
The Markov Widget This widget uses Markov chains to ‘predict’ which component you may want to use next based
on your behavior in the past. A Markov chain is a process that consists of a finite number of states (or levels) and
some known probabilities. It can take some time for this widget to become accustomed to a particular user, but
over time you should begin to notice that this widget will begin to suggest components that you may want to use
next. The Markov Widget can suggest up to five possible components depending on your recent activity. You can
right-click on the Markov widget (the default location is the bottom left-hand corner of the canvas) to dock it into one
of the other corners of the canvas or to hide it completely.
Although a lot of thought has gone into the placement of each component on the component panel to make it
intuitive for new users, people sometimes find it difficult to locate a specific component that might be buried deep
inside one of the category panels. Fortunately, you can also find components by name, by double-clicking on any
empty space on the canvas. This will invoke a pop-up search box. Simply type in the name of the component you
are looking for and you will see a list of parameters or components that match your request.
Double-click anywhere on the canvas to invoke a key word search for a particular component found in the
Component Panels.
A search for “divide” lists a variety of components.
1. Division operator component.
2. Divide Surface component.
1.1.2.8. USING THE SEARCH FEATURE
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22The Grasshopper UI
3. Divide Domain2 component.
There are literally hundreds (if not thousands) of Grasshopper components which are available to you and it can be
daunting as a beginner to know where to look to find a specific component within the Component Palettes. The
quick solution is to double-click anywhere on the canvas to launch a search query for the component you are
looking for. However, what if we need to find a particular component already placed on our canvas? No need to
worry. By right-clicking anywhere on the canvas or pressing the F3 key, you can invoke the Find feature. Start by
typing in the name of the component that you are looking for.
The Find feature employs the use of some very sophisticated algorithms which search not only for any instances of
a component’s name within a definition (a component’s name is the title of the component found under the
Component Panel which we as users cannot change), but also any unique signatures which we may have
designated for a particular component (known as nicknames). The Find feature can also search for any component
type on the canvas or search through text panel, scribble, and group content. Once the Find feature has found a
match, it will automatically grey out the rest of the definition and draw a dashed line around the highlighted
component. If multiple matches are found, a list of components matching your search query will be displayed in the
Find dialog box and hovering over an item in the list will turn that particular component on the canvas green.
By right-clicking anywhere on the canvas or pressing the F3 key, you can invoke the Find feature. Start by
typing in the name of the component that you are looking for.
1.1.2.9. THE FIND FEATURE
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23The Grasshopper UI
The Find feature can be quite helpful to locate a particular component on the canvas. Right-click anywhere on
the canvas to launch the Find dialog box.
A small arrow will also be displayed next to each item in the list which points to its corresponding component
on the canvas. Try moving the Find dialog box around on the canvas and watch the arrows rotate in to keep
track of their components. Clicking on the Find result will try to place the component (on the canvas) next to
the Find dialog box.
As you become more proficient in using the Grasshopper interface, you’ll begin to find ways to expedite your
workflow. Using shortcuts is one way to do this, however there is another feature which can allow you to quickly
access a number of useful tools – the radial UI menu. You can invoke the radial menu by hitting the space bar
(while your mouse is over the canvas or a component) or by clicking your middle mouse button. The radial menu
will enable different tools depending on whether you invoke the menu by clicking directly on top of a component, or
just anywhere on the canvas. In the image below, you see the radial menu has more features available when
clicking on top of a selected component versus just clicking anywhere else on the canvas. This menu can
dramatically increase the speed at which you create Grasshopper documents.
1.1.2.10. USING THE RADIAL MENU
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24The Grasshopper UI
The Radial UI menu allows you to quickly access frequently used menu items.
The canvas toolbar provides quick access to a number of frequently used Grasshopper features. All of the tools are
available through the menu as well, and you can hide the toolbar if you like. The toolbar can be re-enabled from the
View tab on the Main Menu Bar.
1. Open File: A shortcut to open a Grasshopper File.
2. Save File: A shortcut to save the currentGrasshopper File.
3. Zoom Defaults: Default zoom settings that allow you to zoom in or out of your canvas at predefined
intervals.
4. Zoom Extents: Zoom to the extents of your definition. Click on the arrow next to the Zoom Extents icon to
select oneof the sub-menu items to zoom to a particular region within your definition.
5. Named Views: This feature exposes a menu allowing you to store or recall any view area in your
definition.
6. The Sketch Tool: The sketch tool works similarly to the pencil tool set found in Adobe Photoshop with a
few added features.
1.1.2.11. THE CANVAS TOOLBAR
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25The Grasshopper UI
1. Preview Settings: If a Grasshopper component generates some form of geometry, then a preview of this
geometry will be visible in the viewport by default. You can disable the preview on a perobject basis by
right-clicking each component and de-activating the preview feature, or globally change the preview state
by using one of these three buttons.
2. Wire-frame preview.
3. Turn off preview.
4. Shaded preview (default).
5. Preview Selected Objects: With this button toggled, Grasshopper will only display geometry that is part
of selected components, even if those components have a preview=off state.
6. Document Preview Settings: Grasshopper has a default color scheme for selected (semi-transparent
green) and unselected (semi-transparent red) geometry. It is possible to override this color scheme with
the Document Preview Settings dialog.
7. Preview Mesh Quality: For optimization purposes, these settings allow you to control the quality of the
mesh/surface display of the geometry rendered in Rhino. Higher quality settings will cause longer
calculation times, whereas lower settings will display less accurate preview geometry. It should be noted
that the geometry still maintains a high-degree of resolution when baked into the Rhino document –
these settings merely effect the display performance and quality.
The sketch tool allows changes to the line weight, line type, and color. By right-clicking on the selected sketch
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26The Grasshopper UI
object you can choose to simplify your line to create a smoother effect. Right-click on your sketch object and
select “Load from Rhino”. When prompted, select any 2D shape in your Rhino scene. Once you have
selected your referenced shape, hit Enter, and your previous sketch line will be reconfigured to your Rhino
reference shape.
Note: Your sketch object may have moved from its original location once you have loaded a shape from
Rhino. Grasshopper places your sketch object relative to the origin of the canvas (upper left hand corner) and
the world xy plane origin in Rhino.
Grasshopper has a default color scheme for selected (semi-transparent green) and unselected (semi-
transparent red) geometry. It is possible to override this color scheme with the Document Preview Settings
dialog.
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27The Grasshopper UI
1. Grasshopper preview geometry.
2. Rhino viewports.
3. Grasshopper Application window.
All geometry that is generated using the various Grasshopper components will show up (by default) in the Rhino
viewport. This preview is just an Open GL approximation of the actual geometry, and as such you will not be able to
select the geometry in the Rhino viewport (you must first bake it into the scene). You can turn the geometry preview
on/off by right-clicking on a component and selecting the Preview toggle. The geometry in the viewport is color
coded to provide visual feedback. The image below outlines the default color scheme.
Note: This is the default color scheme, which can be modified using the Document Preview Settings tool on
the canvas toolbar.
1.1.3. TALKING TO RHINO
Unlike a Rhino document, a Grasshopper definition does not contain any actual objectsor geometry. Instead, a
Grasshopper definition represents a set of rules & instructions for how Rhino can automate tasks.
1.1.3.1. VIEWPORT FEEDBACK
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1. Green geometry in the viewport belongs to a component which is currently selected.
2. Red geometry in the viewport belongs to a component which is currently unselected.
3. Point geometry is drawn as a cross rather than a rectangle to distinguish it from other Rhino point
objects.
4. Blue feedback means you are currently making a selection in the Rhino Viewport.
Grasshopper is a dynamic environment. Changes that are made are live and their preview display is updated in the
Rhino viewport.
1.1.3.2. LIVE WIRES
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29Talking to Rhino
When storing geometry as internalized in a Grasshopper parameter, the gumball allows you to interface with that
geometry in the Rhino viewport. This update is live and updates will occur as you manipulate the gumball. In
contrast, geometry referenced from Rhino directly will continue to exist in the Rhino document and updates from
Grasshopper will happen only after any changes occur (as opposed to during).
In order to work with (select, edit, transform, etc.) geometry in Rhino that was created in Grasshopper, you must
“bake” it. Baking instantiates new geometry into the Rhino document based on the current state of the Grasshopper
graph. It will no longer be responsive to the changes in your definition.
1.1.3.3. GUMBALL WIDGET
1.1.3.4. BAKING GEOMETRY
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30Talking to Rhino
1. Bake by right-clicking a component and selecting Bake.
2. A dialog will appear that allows you to select onto which Rhino layer the geometry will bake.
3. Grouping your baked geometry is a convenient way to manage the instantiated Rhino geometry,
particularly if you are creating many objects with Grasshopper.
Grasshopper inherits units and tolerances from Rhino. To change the units, type Document Properties in the Rhino
command line to access the Document Properties menu. Select Units to change the units and tolerances.
1.1.3.5. UNITS & TOLERANCES
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Change the units and tolerances in the Rhino Document Properties menu.
Once you get the hang of it, Grasshopper is an incredibly powerful and flexible tool which allows you to explore
design iterations using a graphic interface. However, if you’re working with a single screen then you may have
already noticed that the Grasshopper editor takes up a lot of screen real-estate. Other than constantly zooming in
and out and moving windows around your screen, there really isn’t an elegant solution to this problem. That is…
until the release of the Remote Control Panel!
The Remote Control Panel (RCP) provides a minimal interface to control your definition without taking up a
substantial portion of your screen. The RCP can be instantiated by clicking on the toggle under the View menu of
the Main Menu bar. By default, the RCP is blank — meaning it doesn’t contain any information about your current
Grasshopper document. To populate the RCP with UI elements like sliders, toggles, and buttons, simply right click
on the element and click Publish To Remote Panel. This will create a new group and create a synchronized UI
element in the RCP. Changing the value of the element in the RCP will also update the value in the graph, as well
as modify any geometry in the viewport which might be dependant on this parameter. You can publish multiple
elements and populate a complete interface which can be used to control your file without having the clutter of the
visual graph showing up on top of the Rhino viewport.
Note: The RCP will inherit the UI elements name and use it as the label. It is good practice to update your
sliders and toggles with comprehensible and meaningful names. This will translate directly to your RCP
making it easier to use.
1.1.3.6. REMOTE CONTROL PANEL
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In order to get a UI element (eg. slider, toggle, button, etc.) to show up in the Remote Control Panel, we have
to first publish it.
The RCP UI can also be customized – allowing you to control where objects appear in the interface, the names and
colors of different groups. To modify the layout of the RCP you first have to switch from Working Mode (the default
RCP view) to Edit Mode. You canenter the Editing Mode by clicking on the green pencil in the upper right hand
corner of the RCP. Once in Editing Mode, you can create new UI groups, rearrange elements within groups, add
labels, change colors and more. To delete a UI element, simply drag the element outside the border of the RCP.
You cannot change the individual values of the parameters if you are in Editing Mode. Instead, you will have to click
on the green pencil icon to switch back to the standard Working Mode.
The Remote Control Panel has two modes: Edit Mode (left) which allows you to reorganize the look and feel
of the RCP, and Working Mode where you can modify the actual values of the UI elements.
The Remote Control Panel in Edit Mode has an orange background.
If your Grasshopper file references geometry from Rhino, you must open that same file for the definition to work.
Keep your files organized by storing the Grasshopper and Rhino files in the same folder, and giving them related
names.
1.1.3.7. FILE MANAGEMENT
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1. Project Folder.
2. Rhino file.
3. Grasshopper file.
Creating and specifiying a template file in your Grasshopper preferences is convenient way to set up every new
Grasshopper definition you create. The template can include Grasshopper components as well as panels and
sketch objects for labeling.
1.1.3.8. TEMPLATES
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Create a template file and save it
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35Talking to Rhino
1. In File/Preferences, load the file you just created under Template File. Your template will now be used
each time you create a new file.
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1.2. ANATOMY OF A GRASSHOPPER DEFINTION
Grasshopper allows you to create visual programs called definitions. These
definitions are made up of nodes connected by wires. The following chapter
introduces Grasshopper objects and how to interact with them to start building
definitions.
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37Anatomy of a Grasshopper Definition
Parameters store the data - numbers, colors, geometry, and more - that we send through the graph in our definition.
Parameters are container objects which are usually shown as small rectangular boxes with a single input and
single output. We also know that these are parameters because of the shape of their icon. All parameter objects
have a hexagonal border around their icon.
Geometry parameters can reference geometry from Rhino, or inherit geometry from other components. The point
and curve objects are both geometry parameters.
Input parameters are dynamic interface objects that allow you to interact with your definition. The number slider and
the graph mapper are both input parameters.
Components perform actions based on the inputs they receive. There are manytypes of components for different
tasks.
1. The multiplication component is an operator that calculates the product of twonumbers.
2. The Divide component operates on geometry, dividing a curve into equal segments.
3. The Circle CNR component constructs a circle geometry from input data; a center point, normal vector,
and radius.
4. The Loft component constructs a surface by lofting curves.
We can glean some information about the state of each object based on their color. Let’s take a look at
1.2.1. GRASSHOPPER OBJECT TYPES
Grasshopper consists of two primary types of user objects: parameters andcomponents. Parameters store
data, whereas components perform actions that resultin data. The most basic way to understand Grasshopper
is to remember that we willuse data to define the inputs of actions (which will result in new data that we
cancontinue to use).
1.2.1.1. PARAMETERS
1.2.1.2. COMPONENTS
1.2.1.3. OBJECT COLORS
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Grasshopper’s default color coding system.
A parameter which contains neither warnings nor errors is shown in light gray. This color object indicates that
everything is working properly with this parameter.
A parameter which contains warnings is displayed as an orange box. Any object which fails to collect data is
considered suspect in a Grasshopper definition since it is not contributing to the solution.. Therefore, all
parameters (when freshly added) are orange, to indicate they do not contain any data and have thus no functional
effect on the outcome of the solution. By default, parameters and components that are orange also have a small
balloon at the upper right hand corner of the object. If you hover your mouse over this balloon, it will reveal
information about why the component is giving you a warning. Once a parameter inherits or defines data, it will
become grey and the baloon will disappear.
A component is always a more involved object, since we have to understand and then coordinate what its inputs
and outputs are. Like parameters, a component with warnings is displayed as orange. Remember, warnings aren’t
necessarily bad, it usually just means that Grasshopper is alerting you to a potential problem in your definition.
A component which contains neither warnings nor errors is shown in light gray.
A component whose preview has been disabled is shown in a slightly darker gray. There are two ways to disable a
component’s preview. First, simply right-click on the component and toggle the preview button. To disable the
preview for multiple components at the same time, first select the desired components and then toggle the disable
preview icon (blindfolded man) by right clicking anywhere on the canvas.
A component that has been disabled is shown in a dull gray. To disable a component you may right-click on the
component and toggle the disable button, or you may select the desired components, right click anywhere on the
canvas and select Disable. Disabled components stop sending data to downstream components.
A component which has been selected will be shown in a light green color. If the selected component has
generated some geometry within the Rhino scene, this will also turn green to give you some visual feedback.
A component which contains at least 1 error is displayed in red. The error can come either from the component
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39Grasshopper Object Types
itself or from one of its inputs or outputs.
1. A parameter with no warnings or erros
2. A parameter with warnings
3. A component with warnings
4. A component with no warnings or errors
5. A component with preview disabled
6. A component that has been disabled
7. A selected component
8. A component with an error
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1. The three input parameters of the Circle CNR component.
2. The Circle CNR component area.
3. The output parameter of the Circle CNR component.
A component requires data in order to perform its actions, and it usually comes up with a result. That is why most
components have a set of nested parameters, referred to as Inputs and Outputs, respectively. Input parameters are
positioned along the left side, output parameters along the right side.
There are a few Grasshopper components that have inputs but no outputs, or vice versa. When a component
doesn’t have inputs or outputs, it will have a jagged edge.
Every Grasshopper object has a unique icon. These icons are displayed in the center area of the object and
correspond to the icons displayed in the component palettes. Objects can also be displayed with text labels. To
switch between icon and label display, Select “Draw Icons” from the display menu. You can also select “Draw Full
Names” to display the full name of each object as well as its inputs and outputs.
1.2.2. GRASSHOPPER COMPONENT PARTS
Components are the objects you place on the canvas and connect together with Wiresto form a visual
program. Components can represent Rhino Geometry or operationslike Math Functions. Components have
inputs and outputs.
1.2.2.1. LABEL VS ICON DISPLAY
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41Grasshopper Component Parts
1. Switch betweenIcon and Label display.
2. Display the full name of the component and its inputs and outputs
1. The Circle CNR component in Label Display
2. The Circle CNR component in Icon Display
3. The Circle CNR component with full names displayed
We reccommend using icon display to familiarize yourself with the component icons so you can quickly locate them
in the palettes. This will also enable you to understand definitions at a glance. Text labels can be confusing
because different components may share the same label.
Circle CNR and Circle 3pt have the same label, but different icons.
One feature that can help you familiarize yourself with the location of components in the palettes is holding down
Ctrl + Alt and clicking on an existing component on the canvas. This will reveal its location in the palette.
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Right clicking an object and selecting “Help” from the drop-down menu will open a Grasshopper help window. The
help window contains a more detailled description of the object, a list of inputs and outputs, as well as remarks.
1.2.2.2. COMPONENT HELP
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1. Grasshopper help window for the Point parameter
2. The remarks in the help window give additional insight about the point parameter.
Component inputs are expecting to receive certain types of data, for example a Component might indicate that you
should connect a point or plane to its input. When you hover your mouse over the individual parts of a Component
object, you’ll see different tooltips that indicate the particular type of the sub-object currently under the mouse.
Tooltips are quite informative since they tell you both the type and the data of individual parameters.
1. Header of the tooltip shows the icon for the input type, the name of the Component, the label for the input,
and the input type again in text format.
2. The flain language description of what the input is for the Component.
3. Any values defined for the input - either locally or from its connected wire.
4. The header of the output tooltip provides the same detail os for inputs, but for the corresponding output.
5. The result of the component's action.
All objects on the Canvas have their own context menus that expose their settings and details. You can access this
context menu by right-clicking on the center area of each component. Inputs and outputs each have their own
context menus which can be accessed by right-clicking them.
1.2.2.3. TOOL TIPS
1.2.2.4. CONTEXT POPUP MENUS
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1. Component context menu.
2. Editable text field that lists the name of the object.
3. Preview flag - indicates whether or not the geometry produced by this object will be visible in the Rhino
viewports. Switching off preview will speed up both the Rhino viewport frame-rate and the time taken for a
solution.
4. Runting warnings - lists warnings that are hindering the functioning of the component.
1. C input context menu.
2. Set one or multiple points - allows you to select reference geometry in the Rhino viewport.
3. Manage Point collection - opens a dialog that allows you to add or remove points from the point
collection and view information about each point.
4. Add item to collection.
5. Delete selection.
Some components can be modified to increase the number of inputs or outputs through the Zoomable User
Interface (ZUI). By zooming in on the component on the canvas, an additional set of options will appear which
1.2.2.5. ZOOMABLE USER INTERFACE
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allows you add or remove Inputs or Outputs to that component. The Addition component allows you to add inputs,
representing additional items for the addition operation.
1. Click the + sign to add an Input.
2. Click the - sign to remove an Input.
The panel component also has a zoomable user interface. A Panel is like a Post-It™ sticker. It allows you to add
little remarks or explanations to a Document. You can change the text through the menu or by double-clicking the
panel surface. Panels can also receive and display data from elsewhere. If you plug an output into a Panel, you can
see the contents of that parameter in real-time. All data in Grasshopper can be viewed in this way. When you zoom
in on a panel, a menu appears allowing you to change the background, font, and other attributes. These options are
also available when you right-click the panel
1. Drag grips to adjust panel margins.
2. Increase or reduce the font size of the panel content.
3. Change the alignment of panel content.
4. Select a font for ponel conent.
5. Select a color for the panel background. You can set a new default color for your panels by right clicking
the panel and selecting "Set Defaut Color".
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Persistent data is accessed through the menu, and depending on the kind of parameter has a different manager. A
Point parameter for example allows you to set one or more points through its menu. But, let’s back up a few steps
and see how a Point Parameter behaves.
When you drag and drop a Point Parameter from the Params/Geometry Panel onto the canvas, the Parameter is
orange, indicating it generated a warning. It’s nothing serious, the warning is simply there to inform you that the
parameter is empty (it contains no persistent records and it failed to collect volatile data) and thus has no effect on
the outcome of the solution. The context menu of the Parameter offers two ways of setting persistent data: single
and multiple. Right click on the parameter to set Multiple Points. Once you click on either of these menu items, the
Grasshopper window will disappear and you will be asked to pick a point in one of the Rhino viewports.
Once you have defined all the points you can press Enter and they will become part of the Parameters persistent
data record. This means the Parameter is now no longer empty and it turns from orange to grey. (Notice that the
information balloon in the upper right corner also disappears as there are no more warnings). At this point you can
use the points stored in this Parameter for any subsequent input in your definition.
1. The parameter is orange, indicating it contains no persistent records (and it failed to collect volatile data)
and thus has no effect on the outcome of the solution. Right click on any parameter to set its persistent
data.
2. Once the parameter contains some persistent data, the component will turn from orange to grey.
1.2.3. DATA TYPES
Most parameters can store two different kinds of data: Volatile and Persistent. Volatile data is inherited from
one or more sources and is destroyed (i.e. recollected) whenever a new solution starts. Persistent data is data
which has been specifically set by the user.
1.2.3.1. PERSISTENT DATA
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3. The tooltip for the point parameter shows the persistent data (a collection of referenced points) that is
stored.
Volatile data, as the name suggests, is not permanent and will be destroyed each time the solution is expired.
However, this will often trigger an event to rebuild the solution and update the scene. Generally speaking, most of
the data generated ‘on the fly’ is considered volatile.
As previously stated, Grasshopper data is stored in Parameters (either in Volatile or Persistent form) and is used in
various Components. When data is not stored in the permanent record set of a Parameter, it must be inherited from
elsewhere. Every Parameter (except output parameters) defines where it gets its data from and most Parameters
are not very particular. You can plug a number Parameter (which just means that it is a decimal number) into an
integer source and it will take care of the conversion.
You can change the way data is inherited and stored in the context menu of a parameter or component input. To
change store referenced Rhino geometry in the grasshopperdefinition itself, right click a parameter and select
Internalise data from the menu. This is useful if you want your grasshopper definition to be independent from a
Rhino file.
You can also Internalise data in a component input. Once you select Internalise data in the menu, any wires will
disconnect from that input. The data has been changed from volatile to persistent, and will no longer update.
If you want the data to become volatile again, simply reconnect the wires to the input and the values will
automatically be replaced. You can also right click the input and select Extract parameter. Grasshopper will create a
parameter connected to the input with a wire that contains the data.
1.2.3.2. VOLATILE DATA
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Grasshopper has a variety of Parameters that offer you the ability to interface with the data that is begin supplied to
Component inputs and thereby control for changing the result of your definition. Because they Parameters that
change with our input, they generate Volatile Data.
Number Slider The number slider is the most important and widely used Input Parameter. It allows us to output a
value between two given extremes by interacting with its grip with our mouse. Sliders can be used to specify a value
and see the change to our deifnition that comes with adjusting the grip, but a slider should also be thought of as
the means to identify successful ranges of our definition.
1. Drag the slider grip to change the value - each time you do this, Grasshopper will recompute the
solution.
2. Right click the slider component to change the name, type, and values.
3. Editable text field for the slider name.
4. Select the type of number for the slider to use.
5. Edite the range of values.
6. Double click the name portion of the slider component to open the Slider Editor.
Graph mapper The Graph Mapper is a two-dimensional interface with which we can modify numerical values by
plotting the input along the Graph’s X Axis and outputting the corresponding value along the Y Axis at the X value
intersection of the Graph. It is extremely useful for modulating a set of values within an institutive, grip-based
interface.
1.2.3.3. INPUT PARAMETERS
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1. Move the grips to edit the graph - each time you do this, Grasshopper will recompute the solution.
2. Right click the graph mapper compenent to select the graph type.
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1. Double click the graph mapper to open the Graph Editor.
2. Change thex and y domains.
Value List The Value List stores a collection of values with corresponding list of Labels associated by way of an
equal sign. It is particularly useful when you want to have a few options, labeled with meaning, that can supply
specific output values.
1. Right click the Value List component and select an option from the menu.
2. Double click the Value List component to open the editor and add or change values.
3. In Dropdown List mode, click the arrow to select one of the values. The solution will recompute each
time you change the value.
4. In Check List mode, click next to each value to check it. The component will output all the values that are
checked.
5. In Value Sequence and Value Cycle modes, click the left and right facing arrows to cycle through the
values.
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To connect components, click and drag near the circle on the output side of an object. A connecting wire will be
attached to the mouse. Once the mouse hovers over a potential target input, the wire will connect and become
solid. This is not a permanent connection until you release the mouse button. It doesn’t matter if we make the
connections in a ‘left to right’ or ‘right to left’ manner.
1. The Divide Curve component - divides a curve into equal length segments.
2. Curve parameter - right click and select Set One Curve to reference Rhino Geometry.
Left click and drag the wire from the output (1.) of one object to the input (2.) of another.
1. If you hold down CONTROL, the cursor will become red, and the targeted source will be removed from
the source list.
2. By default, a new connection will erase existing connections. Hold the SHIFT button while dragging
connection wires to difne multiple sources. The cursor will turn green to indicate the addition behavior.
1.2.4. WIRING COMPONENTS
When data is not stored in the permanent record set of a parameter, it must be inherited from elsewhere. Data
is passed from one component to another through wires. You can think of them literally as electrical wires that
carry pulses of data from one object to the next.
1.2.4.1. CONNECTION MANAGEMENT
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1. You can also disconnect wires through the context popup menu - right click the grip of the input or output
and select disconnect.
2. If there are multiple connections, select the one you want to disconnect from the list.
3. When you hover over an item, the wire will be highlighted in red.
Wires represent the connections as well as the flow of data within the graph in our definition. Grasshopper can also
give us visual clues as to what is flowing through the wires. The default setting for these so-called “fancy wires” is
off, so you have to enable them before you can view the different types of line types for the connection wires. To do
this, simply click on the Display Tab on the Main Menu Bar and select the button labeled “Draw Fancy Wires.” Fancy
wires can tell you a lot of information about what type of information is flowing from one component to another.
1. Empty Item – An orange wire type indicates that no information has been transferred. This parameter
has generated a warning message because it contains no data, and thus no information is being sent
across the wire.
2. The Merge component is an alternative to conecting more than one source to a single input.
3. List – If the information flowing out of a component contains a list of information, the wire type will be
shown as a grey double line.
4. Single Item – The data flowing out of any parameter that contains a single item will be shown with a solid
grey line.
1.2.4.2. FANCY WIRES
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5. Tree – Information transferred between components which contain a data structure will be shown in a
grey double-line-dash wire type.
If you have spent any great deal of time working on a single Grasshopper definition, you may have realized that the
canvas can get cluttered with a nest of wires quite quickly. Fortunately, we have the ability to manage the wire
displays for each input of a component.
There are three wire displays: Default Display, Faint Display, and Hidden Display. To change the wire display, simply
right-click on any input on a component and select one of the views available under the Wire Display pop out menu.
1. Hidden Display – When hidden display is selected, the wire will be completely ‘invisible’. The data is
transferred ‘wirelessly’ from the source to the input parameter. If you select the source or target
component, a green wire will appear to show you which components are connected to each other. Once
you deselect the component, the wire will disappear.
2. Default Display – The default wire display will draw all connections (if fancy wires is turned on).
3. Faint Display – The faint wire display will draw the wire connection as a very thin, semi-transparent line.
Faint and Hidden wire displays can be very helpful if you have many source wires coming into a single
input.
1.2.4.3. WIRE DISPLAY
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Grasshopper visual programs are executed from left to right. Reading the graph relative to the wired connections
from upstream to downstream provides understanding about the Program Flow.
Directionalityof data is left to right.
All of the objects and the wires connecting the objects represent the logical graph of our program. This graph
reveals the flow of data, dependencies of any input to its wiired output. Any time our graph changes, sometimes
referrred to as being “dirtied,” every downstream connection and object will be updated.
1. Reparameterize the curve domain between 0.0 and 1.0.
2. Reference a curve from Rhino.
3. Divide the curve into 13 equal segments.
4. Run the parameter values at each curve division point through the graph.
5. Multiply each value by 27.
6. Draw a circle at each division point along th ecurve, normal to the tangent vector at each pont, with a
radius defined by the parameter values (t) modified by the graph mapper and multiplied by 27.
1.2.5. THE GRASSHOPPER DEFINITION
Grasshopper Definitions have a Program Flow that represents where to start program execution, what to do in
the middle and how to know when program execution is complete.
1.2.5.1. PROGRAM FLOW
1.2.5.2. THE LOGICAL PATH
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7. Loft a surface between the circles
1. Variable circle radius.
2. Loft between circles.
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1.3. Building Blocks of Algorithms
This chapter will introduce you to basic geometric and mathematical concepts
and how they are implemented and manipulated in Grasshopper.
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Points in 3D space have three coordinates, usually referred to as [x,y,z]. Points in 2D space have only two
coordinates which are either called [x,y] or [u,v] depending on what kind of two dimensional space we’re talking
about. 2D parameter space is bound to a finite surface. It is still continuous, I.e. hypothetically there are an infinite
amount of points on the surface, but the maximum distance between any of these points is very much limited. 2D
parameter coordinates are only valid if they do not exceed a certain range. In the example drawing, the range has
been set between 0.0 and 1.0 for both [u] and [v] directions, but it could be any finite domain. A point with
1.3.1. Points, Planes & Vectors
Everything begins with points. A point is nothing more than one or more values called coordinates. The number
of coordinate values corresponds with the number of dimensions of the space in which it resides. Points,
planes, and vectors are the base for creating and transforming geometry in Grasshopper.
1.3.1.1 POINTS
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coordinates [1.5, 0.6] would be somewhere outside the surface and thus invalid.
Since the surface which defines this particular parameter space resides in regular 3D world space, we can always
translate a parametric coordinate into a 3D world coordinate. The point [0.2, 0.5] on the surface for example is the
same as point [1.8, 2.0, 4.1] in world coordinates. Once we transform or deform the surface, the 3D coordinates
which correspond with [0.2, 0.5] will change.
If this is a hard concept to grasp, it might help to think of yourself and your position in space. We tend to use local
coordinate systems to describe our whereabouts; “I’m sitting in the third seat on the seventh row in the movie
theatre”, “I’m in the back seat”. If the car you’re in is on the road, your position in global coordinates is changing all
the time, even though you remain in the same back seat ‘coordinate’.
A vector is a geometric quantity describing Direction and Magnitude. Vectors are abstract; ie. they represent a
quantity, not a geometrical element.
Vectors are indistinguishable from points. That is, they are both lists of three numbers so there’s absolutely no way
of telling whether a certain list represents a point or a vector. There is a practical difference though; points are
absolute, vectors are relative. When we treat a list of three doubles as a point it represents a certain coordinate in
space, when we treat it as a vector it represents a certain direction. A vector is an arrow in space which always
starts at the world origin (0.0, 0.0, 0.0) and ends at the specified coordinate.
1.3.1.2. VECTORS
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Planes are “Flat” and extend infinitely in two directions, defining a local coordinate system. Planes are not genuine
objects in Rhino, they are used to define a coordinate system in 3D world space. In fact, it’s best to think of planes
as vectors, they are merely mathematical constructs.
1.3.1.3. PLANES
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1. Attractor point
2. Vectors
3. Circles orient towards attractor based on their normals
In the image above, vectors are drawn between an attractor point and the center point of each circle. These vectors
are used to define the orientation of the circles so they are always facing the attractor point. This same attractor
could be used to change other parameters of the circles. For example, circles that are closest to the attractor could
be scaled larger by using the length of each vector to scale the radius of each circle.
Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
1.3.2. Working with Attractors
Attractors are points that act like virtual magnets - either attracting or repelling other objects. In Grasshopper,
any geometry referenced from Rhino or created withinGrasshopper can be used as an attractor. Attractors can
influence any number of parameters of surrounding objects including scale, rotation, color, and position. These
parameters are changed based on their relationship to the attractor geometry.
1.3.2.1. ATTRACTOR DEFINITION
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In this example, we will use an attractor point to orient a grid of circles, based on the vectors between the center
points of the circles and the attractor point. Each circle will orient such that it is normal to (facing) the attractor point.
01. Type Ctrl+N in Grasshopper to start a new definition
02. Vector/Grid/Hexagonal - Drag and drop the Hexagonal Grid component onto thecanvas
03. Params/Input/Slider - Drag and drop two Numeric Sliders on the canvas
04.
Double-click on the first Numeric Sliders and set the following:
Name: Cell Radius
Rounding: Floating Point
Lower Limit: 0.000
Upper Limit: 1.000
Value: 0.500
05.
Double-click on the second Numeric Sliders and set the following:
Name: # of Cells
Rounding: Integers
Lower Limit: 0
Upper Limit: 10
Value: 10
06. Connect the Number Slider (Cell Radius) to the Size (S) input of the Hexagon Gridcomponent
07. Connect the Number Slider (# of Cells) to the Extent X (Ex) input and the Extent Y (Ey)input of the Hexagon Grid component
08. Curve/Primitive/Circle CNR - Drag and drop a Circle CNR component onto thecanvas
09. Connect the Points (P) output of the Hexagon Grid to the Center (C) input of theCircle CNR component
10. Connect the Number Slider (Cell Radius) to the Radius (R) input of the Circle CNRcomponent.
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11. Vector/Vector/Vector 2Pt - Drag and Drop the Vector 2Pt component onto the
canvas
12. Connect the Points output (P) of the Hexagonal Grid component to the Base Point (A)input of the Vector 2Pt component.
13. Params/Geometry/Point – Drag and Drop the Point component onto the canvas
14. Right-Click the Point component and select set one point. In the model space selectwhere you would like the attractor point to be
15. Connect the Point component to the Tip Point (B) input of the Vector 2Pt component
16. Connect the Vector (V) output of the Vector 2Pt to the Normal (N) input of the CircleCNR component.
17. Curve/Util/Offset – Drag and Drop the Offset Component onto the canvas.
18. Params/Input/Slider - Drag and drop a NumericSlider on the canvas
19.
Double-click on the Number Slider and set the following:
Name: Offset Distance
Rounding: Floating Point
Lower Limit: - 0.500
Upper Limit: 0.500
Value: -0.250
20. Connect the Number Slider (Offset Distance) to the Distance (D) input of the Offsetcomponent
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21. Surface/Freeform/Boundary Surfaces – Drag and drop Boundary Surfaces on tothe canvas
22. Connect the Curves (C) output of the Offset component to the Edges (E) input of theBoundary Surfaces
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
In mathematics, numbers are organized by sets and there are two that you are probably familiar with:
Integer Numbers: […, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …]
Real Numbers: [8, …, -4.8, -3.6, -2.4, -1.2, 0.0, 1.234, e, 3.0, 4.0, …, 8]
While there are other types of number sets, these two interest us the most because Grasshopper uses these
extensively. Although there are limitations in representing these sets exactly in a digital environment, we can
approximate them to a high level of precision. In addition, it should be understood that the distinction between
Integral types (integers) and Floating types (real numbers) corresponds to the distinction between discrete and
continuous domains. In this chapter, we’re going to explore different methods for working with and evaluating
various sets of numbers.
Most of the components that deal with mathematical operations and functions can be found under the following
sub-categories of the Math tab:
1. Domains are used to define a range of values (formerly known as intervals) between two numbers. The
components under the Domain tab allow you to create or decompose different domain types.
2. In mathematics, a matrix is an array of numbers organized in rows and columns. This subcategory
contains a series of utility tools to construct and modify matrices.
3. Operators are used to perform mathematical operations such as Addition, Subtraction, Multiplication, etc.
Conditional operators allow you to determine whether a set of numbers are larger than, less than, or
similar to another set of numbers.
4. Polynomials are one of the most important concepts in algebra and throughout mathematics and
science. You can use the components found in this subcategory to compute factorials, logarithms, or to
raise a number to the nth power.
5. The script subcategory contains single and multi-variable expressions as well as the VB.NET and C#
scripting components.
6. These components allow you to solve trigonometric functions such as Sine,Cosine, Tangent, etc.
7. The time subcategory has a number of components which allow you to construct instances of dates and
times.
8. The utility subcategory is a ‘grab bag’ of useful components that canbe used in various mathematical
equations. Check here if you’re trying find the maximum or minimum values between two lists of
1.3.3. Mathematics, Expressions & Conditionals
Knowing how to work with numeric information is an essential skill to master as you learn to use Grasshopper.
Grasshopper contains many components to perform mathematical operations, evaluate conditions and
manipulate sets of numbers.
1.3.3.1. THE MATH TAB
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numbers; or average a group of numbers.
As was previously mentioned, Operators are a set of components that use algebraic functions with two numeric
input values, which result in one output value.
Most of the time, you will use the Math Operators to perform arithmetical actions on a set of numbers. However,
these operators can also be used on various data types, including points and vectors.
Almost every programming language has a method for evaluating conditional statements. In most cases the
programmer creates a piece of code to ask a simple question of “what if.” What if the area of a floor outline exceeds
the programmatic requirements? Or, what if the curvature of my roof exceeds a realistic amount? These are
important questions that represent a higher level of abstract thought. Computer programs have the ability to analyze
“what if” questions and take actions depending on the answer to that question. Let’s take a look at a very simple
conditional statement that a program might interpret: If the object is a curve, delete it. The piece of code first looks at
an object and determines a single boolean value for whether or not it is a curve. There is no middle ground. The
1.3.3.2. OPERATORS
1.3.3.3. CONDITIONAL OPERATORS
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boolean value is True if the object is a curve, or False if the object is not a curve. The second part of the statement
performs an action dependent on the outcome of the conditional statement; in this case, if the object is a curve then
delete it. This conditional statement is called an If statement. There are four conditional operators (found under the
Math/ Operators subcategory) that evaluate a condition and return a boolean value.
The Equality component takes two lists and compares the first item of List A and compares it to the first item of List
B. If the two values are the same, then a True boolean value is created; conversely if the two values are not equal,
then a False boolean value is created. The component cycles through the lists according to the set data matching
algorithm (default is set to Longest List). There are two outputs for this component. The first returns a list of
boolean values that shows which of the values in the list were equal to one another. The second output returns a
list that shows which values were not equal to one another - or a list that is inverted from the first output.
The Similarity component evaluates two lists of data and tests for similarity between two numbers. It is almost
identical to the way the Equality component compares the two lists, with one exception: it has a percentage input
that defines the ratio of list A that list B is allowed to deviate before inequality is assumed. The Similarity component
also has an output that determines the absolute value distance between the two input lists.
The Larger Than component will take two lists of data and determine if the first item of List A is greater than the first
item of List B. The two outputs allow you to determine if you would like to evaluate the two lists according to a
greater than (>) or greater than and equal to (>=) condition.
The Smaller Than component performs the opposite action of the Larger Than component. The Smaller Than
component determines if list A is less than list B and returns a list of boolean values. Similarly, the two outputs let
you determine if you would like to evaluate each list according to a less than (<) or less than and equal to (<=)
condition.
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
We have already shown that we can use an Expression (or Evaluate) component to evaluate conditional statements
as well as compute algebraic equations. However, there other ways to calculate simple expressions using a few of
the built in Trigonometry functions. We can use these functions to define periodic phenomena like sinusoidal wave
forms such as ocean waves, sound waves, and light waves.
1. Line
 y(t) = 0 
2. Sine Curve
 y(t) = sin(t) 
3. Helix
 x(t) = cos(t) 
 y(t) = sin(t) 
 z(t) = b(t) 
4. Spiral
 x(t) = t*cos(t) 
 y(t) = t*cos(t) 
1.3.3.4. TRIGONOMETRY COMPONENTS
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In this example, we will use Grasshopper to construct various trigonometric curvesusing trigonometry function
components found in the Math tab:
01. Type Ctrl+N (in Grasshopper) to start a new definition
02. Params/Geometry/Point – Drag and drop a Point parameter onto the canvas
03. Right click the Point parameter and click Set One Point – select a point in the Rhinoviewport
04. Vector/Vector/Unit X – Drag and drop the Unit X component to the canvas
05. Params/Input/Number Slider – Drag and drop the Number Slider component ontothe canvas
06.
Double-click on the Number Slider and set the following:
Rounding: Integer
Lower Limit: 10
Upper Limit: 40
Value: 20
07. Transform/Array/Linear Array – Drag and drop the Linear Array component ontothe canvas
08. Connect the output of the Point parameter to the Geometry (G) input of the LinearArray component
09.
Connect the Unit Vector (V) output of the Unit X component to the Direction (D) input
of the Linear Array component
You should see a line of 20 points along the x axis in Rhino. Adjust the slider to
change the number of points in the array.
10. Connect the Number Slider output to the Count (N) input of the Linear ArrayComponent
11. Curve/Spline/Interpolate – Drag and drop the Interpolate Curve component to thecanvas
12. Connect the Geometry (G) output of the Linear Array component to the Vertices (V)input of the Interpolate Curve component
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We have just created a line by connecting an array of points with a curve. Let’s try using some of
Grasshopper’s Trigonometry components to alter this curve:
13. Vector/Point/Deconstruct – Drag and drop a Deconstruct component onto thecanvas
14. Vector/Point/Construct Point - Drag and drop a Construct Point component onto thecanvas
15. Maths/Trig/Sine - Drag and drop a Sine component onto the canvas
16.
Disconnect the wire from the Vertices (V) input of the Interpolate Curve component.
You can disconnect wires by holding down control and dragging, or by right-
clicking the input and selecting Disconnect
17. Connect the Geometry (G) output of the Linear Array component to the Point (P)input of the Deconstruct component
18. Connect the Point X (X) output of the Deconstruct component to the X coordinate (X)input of the Construct Point Component
19. Connect a second wire from the Point X (X) output of the Deconstruct Component tothe Value (x) input of the Sine component
20.
Connect the Result (y) output of the Sine component to the Y coordinate (Y) input of
the Construct Point component
We have now reconstructed our points with the same X values, modifying the Y
values with a sine curve.
21. Connect the Point (Pt) output of the Construct Point component to the Vertices (V)input of the Interpolate component
</li>
You should now see a sine wave curve along the X axis in Rhino
22. Maths/Trig/Cosine – Drag and drop a Cosine component to the canvas
23. Connect a third wire from the Point X (X) output of the Deconstruct Component to theValue (x) input of the Cosine component
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24. Connect the Result (y) output of the Cosine component to the Z coordinate (Z) input
of the Construct Point component
</li>
We have now created a 3D helix
25. Maths/Operators/Multiplication – Drag and drop two Multiplication componentsonto the canvas
26. Connect wires from the Point X (X) output of the Deconstruct component to the (A)input of each Multiplication component
27. Connect the Result (y) output of the Sine component to the (B) input of the firstMultiplication component
28. Connect the Result (y) output of the Cosine component to the (B) input of the secondMultiplication component
29. Disconnect the wire from the Y Coordinate (Y) input of the Construct Pointcomponent
30. Connect the Result (R) output of the first Multiplication component to the XCoordinate (X) input of the Construct Point component
31. Connect the Result (R) output of the second Multiplication component to the ZCoordinate (Z) input of the Construct Point component
You should now see a spiral curve
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
The Expression component (and its brother the Evaluate component) are very flexible tools; that is to say that they
can be used for a variety of different applications. We can use an Expression (or Evaluate component) to solve
mathematical algorithms and return numeric data as the output.
In the following example, we will look at mathematical spirals found in nature and how we can use a few Functions
components to create similar patterns in Grasshopper. We will build on our trigonometric curves definition as a
starting point.
01. Open your Trigonometric curves Grasshopper definition from the previous example
02. Delete the Sine, Cosine, Multiplication, and Interpolate components
03. Params/Input/Number Slider – Drag and drop a Number Slider onto the canvas
04.
Double-click on the Number Slider and set the following:
Rounding: Float
Lower Limit: 0.000
Upper Limit: 1.000
Value: 1.000
05.
Connect the Number Slider to the Factor (F) input of the Unit X component.
This slider allows you to adjust the distance between the points in the array.
06. Maths/Script/Expression – Drag two Expression components onto the canvas
1.3.3.5. EXPRESSIONS
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07. Double-click the first Expression component to open the Expression Editor and
change the expression to: x*sin(x) 
08. Double-click the second Expression component to open the Expression Editor andchange the expression to: x*cos(x) 
Double click the Expression component to open the Grasshopper Expression Editor
09. Connect two wires from the Point X (X) output of the Deconstruct component to theVariable x (x) input of each Expression component
10. Connect the Result (R) output of the first Expression component to the X coordinate(X) input of the Construct Point component
11.
Connect the Result (R) output of the second Expression component to the Y
coordinate (Y) input of the Construct Point component
We have replaced the Trigonometry functions and multiplication operators with
the expression components for a more efficient definition.
12. Mesh/Triangulation/Voronoi – Drag and drop the Voronoi component onto thecanvas
13. Params/Input/Number Slider – Drag and drop a Number Slider onto the canvas
14.
Double-click on the Number Slider and set the following:
Rounding: Integer
Lower Limit: 1
Upper Limit: 30
Value: 30
15. Connect the Number Slider to the Radius (R) input of the Voronoi component
16. Connect the Point (Pt) output of the Construct Point component to the Points (P)input of the Voronoi component
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You can create different Voronoi patterns by manipulating the Factor, Count, and Radius sliders. Below are three
examples:
1. Factor = 1.000, Radius = 15
2. Factor = 0.400, Radius = 10
3. Factor = 0.200, Radius = 7
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
In the color wheel, hue corresponds to the angle. Grasshopper has taken this 0-360 domainand remapped it
between zero and one.
By dividing the Hue domain (0.0 to 1.0) by the number of segments desired, we can assign a hue value to each
segment to create a color wheel.
1.3.4. Domains & Color
The color wheel is a model for organizing colors based on their hue. In Grasshopper, colors can be defined by
their hue value in a range of 0.0 to 1.0. Domains are used to define a range of all possible values between a set
of numbers between a lower limit(A) and an upper limit (B).
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In this example, we will use Grasshopper’s domain and color components to create a color wheel with a variable
amount of segments.
01. Type Ctrl+N (in Grasshopper) to start a new definition
02. Curve/Primitive/Polygon – Drag and drop a Polygon component onto the canvas
03. Params/Geometry/Point – Drag and drop a Point Parameter onto the canvas
04. Right-Click on the Point Component and select set one point
05. Set a point in the model space.
06. Connect the Point Parameter (Base Point) to the Plane (P) input of the Polygoncomponent
07. Params/Input/Number Sliders – Drag and drop two Number Sliders onto the canvas
08.
Double-click on the first Number Sliders and set the following:
Rounding: Integers
Lower Limit: 0
Upper Limit: 10
Value: 10
09.
Double-click on the second Number Sliders and set the following:
Rounding: Integers
Lower Limit: 0
Upper Limit: 100
Value: 37
10.
Connect the Number Slider (Radius) to the Radius (R) input of the Polygon
component
When you connect a number slider to a component in will automatically
change its name to the name of input that it is connecting to.
11. Connect the Number Slider (Segments) to the Segments (S) input of the Polygoncomponent
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12. Curve/Util/Explode – Drag and drop an Explode component onto the canvas.
13. Connect the Polygon (P) output of the Polygon component to the Curve (C) input ofthe Explode component
14. Surface/Freeform/Extrude Point – Drag and drop the Extrude Point component ontothe canvas
15. Connect the Segments (S) output of the Explode component to the Base (B) input ofthe Extrude Point
16. Connect the Point Parameter (Base Point) to the Extrusion Tip (P) of the ExtrudePoint component
17. Surface/Analysis/Deconstruct Brep – Drag and drop the Deconstruct Brepcomponent on to the canvas
18. Connect the Extrusion (E) output of the Extrude Point component to the DeconstructBrep (B) component
19.
Maths/Domain/Divide Domain – Drag and drop the Divide Domain component
The Base Domain (I) is automatically set between 0.0-1.0 which is what we
need for this exercise
20. Connect the Number Slider (Segments) to the Count (C) input of the Divide Domaincomponent
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21. Math/Domain/Deconstruct Domain – Drag and drop the Deconstruct Domaincomponent
22. Connect the Segments (S) output of the Divide Domain component to the Domain (I)input of the Deconstruct Domain component
23. Display/Colour/Colour HSL – Drag and drop the Colour HSL component
24. Connect the Start (S) output of the Deconstruct Domain component to the Hue (H)input of the Colour HSL components
25. Display/Preview/Custom Preview – Drag and drop the Custom Preview component
26.
Right click on the Geometry (G) input of the Custom Preview component and select
Flatten
See 1-4 Designing with Data Trees for details about flattening
27. Connect the Faces (F) output of the Deconstruct Brep component to the Geometry(G) input of the Custom Preview component
28. Connect the Colour (C) output of the Colour HSL component to the Shade (S) input ofthe Custom Preview component
For different color effects, try connecting the Deconstruct Domain component to the saturation (S) or Luminance (L)
inputs of the Colour HSL component.
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Numeric variables can store a whole range of different numbers. Boolean variables can only store two values
referred to as Yes or No, True or False, 1 or 0. Obviously we never use booleans to perform calculations because of
their limited range. We use booleans to evaluate conditions.
Boolean Parameter
In Grasshopper, booleans can be used in several ways. The boolean parameter is a container for one or multiple
boolean values, while the Boolean Toggle allows you to quickly change between single true and false values as
inputs.
Boolean Toggle - double click the boolean value to toggle between true and false
Grasshopper also has objects that test conditions and output boolean values. For example, the Includes
component allows you to test a numeric value to see if it is included in a domain.
The Includes component is testing whether the number 6.8 is included in the domain from 0 to 10. It returns
a boolean value of True.
Logical operators mostly work on booleans and they are indeed very logical. As you will remember, booleans can
only have two values. Boolean mathematics were developed by George Boole (1815-1864) and today they are at
the very core of the entire digital industry. Boolean algebra provides us with tools to analyze, compare and describe
sets of data. Although Boole originally defined six boolean operators we will only discuss three of them:
1. Not
2. And
3. Or
1.3.5. Booleans & Logical Operators
1.3.5.1. BOOLEANS
1.3.5.2. LOGICAL OPERATORS
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The Not operator is a bit of an oddity among operators, because it doesn’t require two values. Instead, it simply
inverts the one on the right. Imagine we have a script which checks for the existence of a bunch of Block definitions
in Rhino. If a block definition does not exist, we want to inform the user and abort the script.
The Grasshopper Not operator (gate)
And and Or take two arguments on either side. The And operator requires both of them to be True in order for it to
evaluate to True. The Or operator is more than happy with a single True value.
As you can see, the problem with Logical operators is not the theory, it’s what happens when you need a lot of them
to evaluate something. Stringing them together quickly results in convoluted code; not to mention operator
precedence problems.
The Grasshopper And operator (gate)
The Grasshopper Or operator (gate)
A good way to exercise your own boolean logic is to use Venn diagrams. A Venn diagram is a graphical
representation of boolean sets, where every region contains a (sub)set of values that share a common property.
The most famous one is the three-circle diagram:
Every circular region contains all values that belong to a set; the top circle for example marks off set {A}. Every value
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inside that circle evaluates True for {A} and every value not in that circle evaluates False for {A}. By coloring the
regions we can mimic boolean evaluation in programming code:
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1.4. Designing with Lists
One of the most powerful features of Grasshopper is the ability to quickly build
and manipulate lists of data. This chapter will explain how to create, manipulate,
and visualize list data.
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Since curves are geometric objects, they possess a number of properties or characteristics which can be used to
describe or analyze them. For example, every curve has a starting coordinate and every curve has an ending
coordinate. When the distance between these two coordinates is zero, the curve is closed. Also, every curve has a
number of control-points, if all these points are located in the same plane, the curve as a whole is planar. Some
properties apply to the curve as a whole, while others only apply to specific points on the curve. For example,
planarity is a global property while tangent vectors are a local property. Also, some properties only apply to some
curve types. So far we’ve discussed some of Grasshopper’s Primitive Curve Components such as: lines, circles,
ellipses, and arcs.
1. Line
2. Polyline
3. Circle
4. Ellipse
5. Arc
6. NURBS Curve
7. Polycurve
1. End Point
2. Edit Point
3. Control Point
Degree: Thedegree is a positive whole number. This number is usually 1, 2, 3 or 5, but can be any positive whole
number. The degree of the curve determines the range of influence the control points have on a curve; where the
1.4.1. CURVE GEOMETRY
NURBS (non-uniform rational B-splines) are mathematical representations that can accurately model any
shape from a simple 2D line, circle, arc, or box to the most complex 3D free-form organic surface or solid.
Because of their flexibility and accuracy, NURBS models can be used in any process from illustration and
animation to manufacturing.
1.4.1.1. NURBS CURVES
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higher the degree, the larger the range. NURBS lines and polylines are usually degree 1, NURBS circles are
degree 2, and most free-form curves are degree 3 or 5.
Control Points: The control points are a list of at least degree+1 points. One of the easiest ways to change the
shape of a NURBS curve is to move its control points.
Weight: Control points have an associated number called a weight . Weights are usually positive numbers. When a
curve’s control points all have the same weight (usually 1), the curve is called non-rational, otherwise the curve is
called rational. Most NURBS curves are non-rational. A few NURBS curves, such as circles and ellipses, are always
rational.
Knots: Knots are a list of (degree+N-1) numbers, where N is the number of control points.
Edit Points: Points on the curve evaluated at knot averages. Edit points are like control points except they are always
located on the curve and moving one edit point generally changes the shape of the entire curve (moving one control
point only changes the shape of the curve locally). Edit points are useful when you need a point on the interior of a
curve to pass exactly through a certain location.
NURBS curve knots as a result of varying degree:
A D1 NURBS curve behaves the same as a polyline. A D1 curve has a knot for every control point.
D2 NURBS curves are typically only used to approximate arcs and circles. The spline intersects with the
control polygon halfway each segment.
D3 is the most common type of NURBS curve and is the default in Rhino. You are probably very familiar with
the visual progression of the spline, even though the knots appear to be in odd locations.
Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
1.4.1.2. GRASSHOPPER SPLINE COMPONENTS
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Grasshopper has a set of tools to express Rhino’s more advanced curve types like nurbs curves and poly curves.
These tools can be found in the Curve/Splines tab.
Nurbs Curve (Curve/Spline/Nurbs curve): The Nurbs Curve component constructs a NURBS curve from control
points. The V input defines these points, which can be described implicitly by selecting points from within the Rhino
scene, or by inheriting volatile data from other components. The Nurbs Curve-D input sets the degree of the curve.
Interpolate Curve (Curve/Spline/Interpolate): Interpolated curves behave slightly differently than NURBS curves. The
V-input is for the component is similar to the NURBS component, in that it asks for a specific set of points to create
the curve. However, with the Interpolated Curve method, the resultant curve will actually pass through these points,
regardless of the curve degree. In the NURBS curve component, we could only achieve this when the curve degree
was set to one. Also, like the NURBS curve component, the D input defines the degree of the resultant curve.
However, with this method, it only takes odd numbered values for the degree input. Again, the P-input determines if
the curve is Periodic. You will begin to see a bit of a pattern in the outputs for many of the curve components, in that,
the C, L, and D outputs generally specify the resultant curve, the length, and the curve domain respectively.
Kinky Curve (Curve/Spline/Kinky Curve): The kinky curve component allows you the ability to control a specific angle
threshold, A, where the curve will transition from a kinked line, to a smooth, interpolated curve. It should be noted
that the A-input requires an input in radians.
Polyline (Curve/Spline/Polyline): A polyline is a collection of line segments connecting two or more points, the
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resultant line will always pass through its control points; similar to an Interpolated Curve. Like the curve types
mentioned above, the V-input of the Polyline component specifies a set of points that will define the boundaries of
each line segment that make up the polyline. The C-input of the component defines whether or not the polyline is
an open or closed curve. If the first point location does not coincide with the last point location, a line segment will
be created to close the loop. The output for the Polyline component is different than that of the previous examples,
in that the only resultant is the curve itself.
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Grasshopper generally has two types of data: persistent and volatile. Even though the data types have different
characteristics, typically Grasshopper stores this data in an array, a list of variables.
When storing data in a list, it’s helpful to know the position of each item in that list so that we can begin to access or
manipulate certain items. The position of an item in the list is called its index number.
1. List Item
2. Index
The only thing that might seem odd at first is that the first index number of a list is always 0; not 1. So, when we talk
about the first item of a list, we actually mean the item that corresponds to index number 0.
For example, if we were to count the number of fingers we have on our right hand, chances are that you would have
counted from 1 to 5. However, if this list has been stored in an array, then our list would have counted from 0 to 4.
Note, that we still have 5 items in the list; it’s just that the array is using a zero-based counting system. The items
being stored in the list don’t just have to be numbers. They can be any data type that Grasshopper supports, such
as points, curves, surfaces, meshes, etc.
Often times the easiest way to take a look at the type of data stored in a list is to connect a Text Panel
(Params/Input/Panel) to the output of a particular component. By default, the Text Panel automatically shows all
index numbers to the left side of the panel and displays the data items on the right side of the panel. The index
numbers will become a crucial element when we begin working with our lists. You can turn the index numbers on
and off by right-clicking on the Text Panel and clicking on the “Draw Indices” item in the sub-menu. For now, let’s
leave the entry numbers turned on for all of our text panels.
1.4.2. What is a List?
It’s helpful to think of Grasshopper in terms of flow, since the graphical interface is designed to have data flow
into and out of specific types of components. However, it is the data that define the information flowing in and
out of the components. Understanding how to manipulate list data is critical to understanding the Grasshopper
plug-in.
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Imagine a component which creates line segments between points. It will have two input parameters which both
supply point coordinates (List A and List B):
As you can see there are different ways in which we can draw lines between these sets of points. New to
Grasshopper 0.9 are three components for data matching, found under the Sets/List panel: Shortest List, Longest
List, and Cross Reference. These new components allow for greater flexibility within the three basic data matching
algorithms. Right clicking each component allows you to select a data matchingoption from the menu.
The simplest way is to connect the inputs one-on-one until one of the streams runs dry. This is called the “Shortest
List” algorithm:
Select a matching algorithm option from the component menu by right-clicking the component.
The “Longest List” algorithm keeps connecting inputs until all streams run dry. This is the default behavior for
1.4.3. Data Stream Matching
Data matching is a problem without a clean solution. It occurs when a component has access to differently
sized inputs. Changing the data matching algorithm can lead to vastly different results.
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components:
Finally, the “Cross Reference” method makes all possible connections:
This is potentially dangerous since the amount of output can be humongous. The problem becomes more intricate
as more input parameters are involved and when the volatile data inheritance starts to multiply data, but the logic
remains the same.
Let’s look more closely at the Shortest List component:
Here we have two input lists {A,B,C,D,E} and {X,Y,Z}. Using the Trim End option, the last two items in the first list are
disregarded., so that the lists are of equal length.
Using the Trim Start option, the first two items in the first list are disregarded, so that the lists are of equal length.
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The Interpolate option skips the second and fourth items in the first list. Now let’s look at the Cross Reference
component:
Here we have two input lists {A,B,C} and {X,Y,Z}. Normally Grasshopper would iterate over these lists and only
consider the combinations {A,X}, {B,Y} and {C,Z}. There are however six more combinations that are not typically
considered, to wit: {A,Y}, {A,Z}, {B,X}, {B,Z}, {C,X} and {C,Y}. As you can see the output of the Cross Reference
component is such that all nine permutations are indeed present.
We can denote the behaviour of data cross referencing using a table. The rows represent the first list of items, the
columns the second. If we create all possible permutations, the table will have a dot in every single cell, as every
cell represents a unique combination of two source list indices.
Sometimes however you don’t want all possible permutations. Sometimes you wish to exclude certain areas
because they would result in meaningless or invalid computations. A common exclusion principle is to ignore all
cells that are on the diagonal of the table. The image above shows a ‘holistic’ matching, whereas the ‘diagonal’
option (available from the Cross Reference]component menu has gaps for {0,0}, {1,1}, {2,2} and {3,3}. If we apply
this to our {A,B,C}, {X,Y,Z} example, we should expect to not see the combinations for {A,X}, {B,Y} and {C,Z}:
The rule that is applied to ‘diagonal’ matching is: “Skip all permutations where all items have the same list index”.
‘Coincident’ matching is the same as ‘diagonal’ matching in the case of two input lists, but the rule is subtly
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different: “Skip all permutations where any two items have the same list index”.
The four remaining matching algorithms are all variations on the same theme. ‘Lower triangle’ matching applies
the rule: “Skip all permutations where the index of an item is less than the index of the item in the next list”, resulting
in an empty triangle but with items on the diagonal.
‘Lower triangle (strict)’ matching goes one step further and also eliminates the items on the diagonal:
‘Upper Triangle’ and ‘Upper Triangle (strict)’ are mirror images of the previous two algorithms, resulting in empty
triangles on the other side of the diagonal line.
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Perhaps the easiest way to create a list (and one of the most over-looked methods) is to manually type in a list of
values into a parameter. Using this method puts added responsibility on the user because this method relies on
direct user input (ie. persistent data) for the list creation. In order to change the list values, the user has to manually
type in each individual valuewhich can be difficult if the list has many entries. There are several ways to manually
create a list. One way is to use a Number paramter. Right click the Number parameter and select “Manage Number
Collection.”
1. Right click the number component to open the Number collection Manager.
2. Click the Add Item icon to add a number to the list.
3. Double click the number to change its value.
Another method is to manually enter the list items into a panel. Make sure that “Multiline Data” is deselected.
1.4.4. Creating Lists
There are many different ways to generate lists in Grasshopper. Below, we’ll look at a few different methods for
generating lists and then look at how the data can be used to convey information in the viewport via a
visualization.
1.4.4.1. MANUAL LIST CREATION
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The Range component, found under Sets/Sequence/Range, creates a list of evenly spaced numbers between a
low and a high value called the Domain. A domain (also sometimes referred to as an interval) is every possible
number between two numeric extremes.
A Range component divides a numeric domain into even segments and returns a list of values.
1. Number of Steps = 10
2. Domain goes from 0 to 1
3. Total number of points = 11
In the example below, the numeric domain has been defined as every possible number between 0 and 20. The
Range component takes that domain and divides it up by the number of steps (in this case 10). So, we have 10
even spaced segments. The Range component returns a list of values. Because it keeps the first and the last
values in the list, the output of a Range component is always one more than the number of steps. In the example
above, we created 10 steps, so the Range component returns 11 values.
Create a list using the Range component by specifying a Domain and number of steps.
You may have noticed something a little quirky about the setup we just made. We know that a domain is always
defined by two values (a high and low value). Yet, in our definition we simply connected a single value to the domain
input. In order to avoid errors, Grasshopper makes an assumption that you are trying to define a domain between
zero and some other number (our slider value). In order to create a range between two numbers that doesn’t start
at zero, we must use the Construct Domain component to specify the domain.
1.4.4.2. RANGE
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To create a Range from a domain that does not start at zero, use the Construct Domain component.
The Series component is similar to the Range component, in that, it also creates a list of numbers. However a
Series component is different because it creates a set of discreet numbers based on a start value, step size, and
the number of values in the series.
The Series component creates a list based on a start value, step value, and the number of values in the list.
The Random Component (Sets/Sequence/Random) can be used to generate a list of pseudo random numbers.
They are referred to as “pseudo” random because the number sequence is unique but stable for each seed value.
Thus, you can generate an entirely new set of random numbers by changing the seed value (S-input). The domain,
as in the previous example, is a defined interval between two numeric extremes.
1.4.4.3. SERIES
1.4.4.4. RANDOM
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
There are many different ways to visualize a list of data. The most common way is to create some geometry with the
list of data. By connecting the R output of the Range component tothe Y input of the Construct Point component, we
can see an array of points in the Y direction.
Lets look at some components that can help us understand the data.
The Point List component is an extremely useful tool for visualizing the order of a set of points in a list. Essentially,
the Point List component places the index item number next to the point geometry in the viewport. You can also
specify whether or not you want to draw the number tags, the connection lines, or the size of the text tags.
You can visualize the order of a set of points using the Point List component.
The text tag component allows you to draw little strings (a string is a set of ASCII characters) in the viewport as
feedback items. Text and location are specified as input parameters. When text tags are baked into the scene, they
turn into Text Dots. The other interesting thing about Text Tags is that they are viewport independent - meaning the
tags always face the camera (including perspective views) and they always remain the same size on the screen
regardless of your zoom settings.
1.4.5. List Visualization
Understanding lists in Grasshopper can be difficult without being able to see the data flowing from one
component to the next. There are several ways to visualize lists that can help to understand and manipulate
data.
1.4.5.1. THE POINT LIST COMPONENT
1.4.5.2. TEXT TAGS
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You can visualize any string information in the viewport using the Text Tag component. In this setup, we have
decided to display the value of each point on top of each point locaiton. We could have assigned any text to
display.
The Text Tag 3d component works very similarly to the Text Tag component. They differ, in that, when Text Tag 3d
objects are baked into the scene, they become Text objects in Rhino. The scale of the Text Tag 3d font can also be
controlled via an input (which is inaccessible in the Text Tag component).
You can use a Text Tag 3d component to visualize information like a Text object in Rhino.
One of the other things we can do to visualize the list data is to assign color to the geometry. Grasshopper has
limited ‘rendering’ capabilities, but we can control simple Open GL settings like color, specular color, transparency,
etc. The L0 value represents the low end (left side) of the gradient, whereas the L1 value represents the upper end
(right side). These values correspond to the start and end of our domain. The t-values are the elements in the list
that will get mapped somewhere within the L0 and L1 range. The output of the gradient is a list of RGB color values
which correspond to each point in our list. Right-click on the Gradient to set one of the gradient presets, or define
your own using the color node points.
1.4.5.3. COLOR
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1. Points
2. Point list
3. Text Tag
4. Text Tag 3D
5. Custom color preview
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
The List Length component (Sets/List/List Length) essentially measures the length of the List. Because our lists
always start at zero, the highest possible index in a list equals the length of the list minus one. In this example, we
have connected our base List to the List Length-L input, showing that there are 6 values in the list.
Our List is fed into a List Item component (Sets/List/List Item) in order to retrieve a specific data item from within a
data set. When accessing individual items in a list, we have to specify the i-input; which corresponds to the index
number we would like to retrieve. We can feed a single integer or a list of integers into the i-input depending on how
many items we would like to retrieve.The L-input defines the base list which we will be analyzing. In this example,
we have set the i-input to 2 so the List Item component returns the data item associated with the 3rd entry number
in our list.
We can invert the order of our list by using a Reverse List component (Sets/List/Reverse). If we input an ascending
list of numbers from 0.0 to 50.0 into the Reverse List component; the output returns a descending list from 50.0 to
0.0.
The Shift List component (Sets/Sequence/Shift List) will either move the list up or down a number of increments
depending on the value of the shift offset. We have connected the List output into the Shift-L input, while also
connecting a number to the Shift-S input. If we set the offset to -1, all values of the list will move down by one entry
1.4.6. List Management
One of the most powerful features of Grasshopper is the ability to quickly build and manipulate various lists of
data. We can store many different types of data in a list (numbers, points, vectors, curves, surfaces, breps,
etc.) and there are a number of useful tools found under the Sets/List subcategory.
1.4.6.1. LIST LENGTH
1.4.6.2. LIST ITEM
1.4.6.3. REVERSE LIST
1.4.6.4. SHIFT LIST
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number. Likewise, if we change the offset to +1, all values of the list will move up by one entry number. If Wrap input
equals True, then items that fall off the ends are re-appended to the beginning or end of the list. In this example, we
have a shift offset value set to +1, so that our list moves up by one entry number. Now, we have a decision to make
on how we would like to treat the first value. If we set the Wrap value to False, the first entry will be shifted up and out
of the list, essentially removing this value from the data set (so, the list length is one less than it was before).
However, if we set the wrap value to True, the first entry will be moved to the bottom of the list
The Insert Items component (Sets/Lists/Insert Items) enables you to insert a collection of items into a list. In order
for this to work properly, you need to know the items you want to insert and the index position for each new item. In
the example below, we will insert the letters A, B, and C into index position three.
The Weave component (Sets/Lists/Weave) merges two or more lists together based on a specified weave pattern
(P input).When the pattern and the streams do not match perfectly, this component can either insert nulls into the
output streams or it can ignore streams which have already been depleted.
1.4.6.5. INSERT ITEMS
1.4.6.6. WEAVE
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The Cull component (Sets/Sequence/Cull Pattern) removes elements in a list using a repeating bit mask. The bit
mask is defined as a list of Boolean (true or false) values. The bit mask is repeated until all elements in the data
list have been evaluated.
1.4.6.7. CULL PATTERN
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Lets take a look at an example using the components from the previous section. In this example, we are creating a
tile pattern by mapping geometry to a rectangular grid. The pattern is created by using the List Item component to
retrieve the desired tile from a list of geometry.
1. Geometry corresponding to index 1
2. Geometry corresponding to index 0
3. Rectangular grid
1. Mapping pattern
2. Mapped geometry
01. Start a Rhinoceros File.
02. Create two equally sized squares.
1.4.7. WORKING WITH LISTS
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03.
Create different geometries in each square.
In the example shown above, we created a simple surface with a tab. The tab
is filleted to demonstrate the orientation and the base is filleted to distinguish
the two geometries.
04. Start a new definition, type Ctrl+N (in Grasshopper).
05. Params/Geometry/Geometry – Drag and drop two Geometry parameters onto thecanvas.
06. Right-Clickthe first Geometry Parameter and select set one Geometry. Set the firstGeometry that you are referencing.
07.
Right-Click the second Geometry Parameter and select set one Geometry. Set the
second Geometry that you are referencing. 
It is possible to reference multiple geometries in a single parameter, but for
simplicity were are using two separate parameter components.
08. Params/Geometry/Curve – Drag and drop two Curve parameters onto the canvas.
09. Right-Click the first Curve Parameter and select set one Curve. Set the first squarethat you are referencing.
10.
Right-Click the second Curve Parameter and select set one Curve. Set the second
square that you are referencing. 
Be sure that the geometry and the square that you are referencing correspond.
11. Vector/Grid/Rectangular – Drag and drop a Rectangular Grid component onto thecanvas.
12. Params/Input/Slider - Drag and drop three Number Sliders on the canvas.
13.
Double-click on the first Number Slider and set the following:
Rounding: Integers
Lower Limit: 0
Upper Limit: 10
Value: 10
14.
Double-click on the second Number Slider and set the following:
Rounding: Integers
Lower Limit: 0
Upper Limit: 10
Value: 10
15.
Double-click on the third Number Slider and set the following:
Name: Extents X & Y
Rounding: Integers
Lower Limit: 0
Upper Limit: 10
Value: 10
16. Connect the first Number Slider to the Size X (Sx) input of the Rectangular Gridcomponent.
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17. Connect the second Number Slider to the Size Y (Sy) input of the Rectangular Grid
component.
18. Connect the third Number Slider to the Extent X (Ex) input and the Extent Y (Ey) inputof the Rectangular Grid component.
19. Sets/Tree/Merge – Drag and drop two Merge components onto the canvas.
20. Connect the first Geometry parameter to Data Stream 1 (D1) input of the first Mergecomponent.
21. Connect the second Geometry parameter to Data Stream 2 (D2) input of the firstMerge component.
22. Connect the first Curve parameter to Data Stream 1 (D1) input of the second Mergecomponent.
23. Connect the second Curve parameter to Data Stream 1 (D2) input of the secondMerge component.
24. Right-click the Cells (C) output of the Rectangular Grid component and selectFlatten.
25. Sets/List/List Length – Drag and drop a List Length component onto the canvas.
26. Connect the Cells (C) output of the Rectangular Grid component to the List (L) inputof the List Length component.
27. Sets/Sequence/Repeat Data – Drag and drop a Repeat Data component onto thecanvas.
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28. Connect the Length (L) output of the List Length component to the Length (L) input of
the Repeat Data component.
29. Params/Input/Panel – Drag and drop a Panel onto the canvas.
30.
Double-click the Panel. Deselect multiline data, wrap items, and special codes.
Enter the following:
1
0
0
This is the pattern in which the geometries are being distributed. 0 is calling
out the first referenced Geometry and 1 is calling out the second referenced
Geometry. Changing the number sequence will change the pattern, as will
changing the extents of the grid.
31. Connect the Panel to the Data (D) input of the Repeat Data component.
32. Sets/List/List Item – Drag and drop two List Item components.
33. Connect the Result (R) output of the first Merge component to the List (L) input of thefirst List Itemcomponent.
34. Connect the Result (R) output of the second Merge component to the List (L) input ofthe second List Item component.
35. Connect the Data (D) output of the Repeat Data component to the Index (i) input ofthe first and second List Item components.
36. Transform/Affine/Rectangle Mapping – Drag and Drop the Rectangle Mappingcomponent onto the canvas.
37. Connect the Cells (C) output of the Rectangular Grid component to the Target (T)input of the Rectangular Mapping component.
38. Connect the items (I) output of the first List Item component to the Geometry (G)input of the Rectangular Mapping component.
39. Connect the items (I) output of the second List Item component to the Source (S)input of the Rectangular Mapping component.
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Changing the input geometry and the pattern will change the final tile pattern.
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1.5. DESIGNING WITH DATA TREES
As your definitions increase in complexity, the amount of data flowing through
also increases. In order to effectively use Grasshopper, it is important to
understand how large quantities of data are stored, accessed, and manipulated.
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Apart from a few primitive surface types such as spheres, cones, planes and cylinders, Rhino supports three kinds
of freeform surface types, the most useful of which is the NURBS surface. Similar to curves, all possible surface
shapes can be represented by a NURBS surface, and this is the default fall-back in Rhino. It is also by far the most
useful surface definition and the one we will be focusing on.
1. Sphere Primitive [plane, radius]
2. Cylinder Primitive [plane, radius, height]
3. Plane Primitive [plane, width, height]
4. Cone Primitive [plane, radius, height]
NURBS surfaces are very similar to NURBS curves. The same algorithms are used to calculate shape, normals,
tangents, curvatures and other properties, but there are some distinct differences. For example, curves have
tangent vectors and normal planes, whereas surfaces have normal vectors and tangent planes.This means that
curves lack orientation while surfaces lack direction. In the case of NURBS surfaces, there are in fact two directions
implied by the geometry, because NURBS surfaces are rectangular grids of {u} and {v} curves. And even though
these directions are often arbitrary, we end up using them anyway because they make life so much easier for us.
You can think of NURBS surfaces as a grid of NURBS curves that go in two directions. The shape of a
NURBS surface is defined by a number of control points and the degree of that surface in the u and v
directions. NURBS surfaces are efficient for storing and representing free-form surfaces with a high degree
of accuracy.
Surface Domain A surface domain is defined as the range of (u,v) parameters that evaluate into a 3-D point on that
surface. The domain in each dimension (u or v) is usually described as two real numbers (u_min to u_max) and
(v_min to v_max) Changing a surface domain is referred to as reparameterizing the surface.
1.5.1. Surface Geometry
NURBS (non-uniform rational B-splines) are mathematical representations that can accurately model any
shape from a simple 2D line, circle, arc, or box to the most complex 3D free-form organic surface or solid.
Because of their flexibility and accuracy, NURBS models can be used in any process from illustration and
animation to manufacturing.
1.5.1.1. NURBS SURFACES
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In Grasshopper, it is often useful to reparameterize NURBS surfaces so that the u and v domains both range from 0
to 1. This allows us to easily evaluate and operate on the surface.
Evaluating parameters at equal intervals in the 2-D parameter rectangle does not necessarily translate into
equal intervals in 3-D space.
Surface evaluation Evaluating a surface at a parameter that is within the surface domain results in a point that is
on the surface. Keep in mind that the middle of the domain (mid-u, mid-v) might not necessarily evaluate to the
middle point of the 3D surface. Also, evaluating u- and v-values that are outside the surface domain will not give a
useful result.
Normal Vectors and Tangent Planes The tangent plane to a surface at a given point is the plane that touches the
surface at that point. The z-direction of the tangent plane represents the normal direction of the surface at that point.
Grasshopper handles NURBS surfaces similarly tothe way that Rhino does because it is built on the same core of
operations needed to generate the surface. However, because Grasshopper is displaying the surface on top of the
Rhino viewport (which is why you can’t really select any of the geometry created through Grasshopper in the
viewport until you bake the results into the scene) some of the mesh settings are slightly lower in order to keep the
speed of the Grasshopper results fairly high. You may notice some faceting in your surface meshes, but this is to
be expected and is only a result of Grasshopper’s drawing settings. Any baked geometry will still use the higher
mesh settings.
In the previous section, we explained that NURBS surfaces contain their own coordinate space desfined by u and v
domains. This means that two dimensional geometry that is defined by x and y coordinates can be mapped onto
the uv space of a surface. The geometry will stretch and change in response to the curvature of the surface. This is
different from simply projecting 2d geometry onto a surface, where vectors are drawn from the 2d geometry in a
specified direction until they intersect with the surface.
1.5.1.2. PROJECTING SURFACES
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You can think of projecting as geometry casting a shadow onto a surface, and mapping as geometry being
stretched over a surface.
1. Mapped geometry defined by uv coordinates
2. Projecting geometry onto a surface
Just as 2d geometry can be projected onto the uv space of a surface, 3d geometry that is contained by a box can be
mapped to a corresponding twisted box on a surface patch. This operation is called box morphing and is useful for
populating curved surfaces with three dimensional geometric components.
To array twisted boxes on a surface, the surface domain must be divided to create a grid of surface patches. The
twisted boxes are created by drawing normal vectors at the corners of each patch to the desired height and creating
a box defined by the end points of those vectors and the corner points of the patch.
1.5.1.3. MORPHING DEFINITION
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
In this example, we will use the box morph component to populate a NURBS surface with a geometric component.
1. NURBS surface populated with component.
2. Original component in reference box.
3. Surface divided into patches.
4. Twisted boxes arrayed on surface.
01. Start a new definition, type Ctrl+N (in Grasshopper)
02.
Params/Geometry/Surface – Drag and drop a Surface parameter onto the canvas
This is the surface that we will populate with geometric components.
03.
Params/Geometry/Geometry – Drag a Geometry parameter to the canvas
This is the component that will be arrayed over the surface.
04. Right click the Surface Parameter and select “Set One Surface” – select a surface toreference in the Rhino viewport
05. Right click the Geometry parameter and select “Set One Geometry” – select the yourRhino geometry
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06. Maths/Domain/Divide Domain2 – Drag and drop the Divide Domain2 componentonto the canvas
07. Params/Input/Number Slider – Drag three Number Sliders onto the canvas
08.
Double click the first Number Slider and set the following:
Rounding: Integer
Lower Limit: 0
Upper Limit: 10
Value: 5
09. Set the same values on the second and third Number Sliders
10. Connect the output of the Surface parameter to the Domain (I) input of the DivideDomain2 component
11. Connect the first Number Slider to the U Count (U) input of the Divide Domain2component
12. Connect the second Number Slider to the V Count (V) input of the Divide Domain2component
13. Transform/Morph/Surface Box – Drag the Surface Box component to the canvas
14. Connect the output of the Surface parameter to the Surface (S) input of the SurfaceBox component
15. Connect the Segements (S) output of the Divide Domain2 component to the Domain(D) input of the Surface Box component
You should see a grid of twisted boxes populating your referenced surface. Change the U and V count sliders
to change the number of boxes, and use the height slider to adjust their height.
16. Connect the third Number Slider to the Height (H) input of the Surface Boxcomponent
17. Surface/Primitive/Bounding Box – Drag a Bounding Box component to the canvas
18. Transform/Morph/Box Morph – Drag and drop the Box Morph component onto thecanvas
19. Connect the output of the Geometry parameter to the Content (C) input of theBounding Box component
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20. Connect the output of the Geometry parameter to the Geometry (G) input of the BoxMorph component
21. Connect the Box (B) output of the Bounding Box component to the Reference (R)input of the Box Morph component
22. Connect the Twisted Box (B) output of the Surface Box component to the Target (T)input of the Box Morph component
You should now see your geometry populating your surface.
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It’s possible to have multiple lists of data inside a single parameter. Since multiple lists are available, there needs
to be a way to identify each individual list. A Data Tree is essentially a list of lists, or sometimes a list of lists of lists
(and so on).
In the image above, there is a single master branch (you could call this a trunk, but since it’s possible to have
multiple master branches, it might be a bit of a misnomer) at path {0}. This path contains no data, but does have 6
sub-branches. Each of these sub-branches inherit the index of the parent branch {0} and add their own sub-index
(0, 1, 2, 3, 4, and 5 respectively). It would be wrong to call this an “index”, because that implies just a single number.
It is probably better to refer to this as a “path”, since it resembles a folder-structure on the disk. At each of these
sub-branches, we encounter some data. Each data item is thus part of one (and only one) branch in the tree, and
each item has an index that specifies its location within the branch. Each branch has a path that specifies its
location within the tree.
The image below illustrates the difference between a list and a data tree. On the left, an array of four columns of six
points each is all contained in one list. The first column numbered 0-5, the second 6-11, and so on. On the right is
the same array of points contained in a data tree. The data tree is a list of four columns, and each column is a list of
six points. The index of each point is (column number, row number). This is a much more useful way of organizing
this data, because you can easily access and operate on all the points in a given row or column, delete every
second row of points, connect alternating points, etc.
1.5.2. What is a Data Tree?
A Data Tree is a hierarchical structure for storing data in nested lists. Data trees are created when a
grasshopper component is structured to take in a data set and output multiple sets of data. Grasshopper
handles this new data by nesting it in the form of sub-lists. These nested sub-lists work in the same way as
folder structures on your computer in that accessing indexed items require moving through paths that are
informed by their generation of parent lists and their own sub-index.
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Due to their complexity, Data Trees can be difficult to understand. Grasshopper has several tools to help visualize
and understand the data stored in a tree.
The Param Viewer The Param Viewer (Params/Util/Param Viewer) allows you to visualize data in text form and as
a tree. Connect any output containing data to the input of the Param Viewer. To show the tree, right-click the Param
Viewerand select “draw tree.” In this example, the Param Viewer is connected to the Points (P) output of a Divide
Curve component that divided 10 curves into 10 segements each. The ten branches correspond to the ten curves,
each containing a list of 11 points which are the division points of the curve.
1. Path of each list
2. Number of items in each list
3. Select "Draw Tree" to display the data tree
If we connect a panel to the same output, it displays ten lists of 11 items each. You can see that each item is a point
defined by three coordinates. The path is displayed at the top of each list, and corresponds to the paths listed in the
Param Viewer.
1.5.2.1. DATA TREE VISUALIZATION
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1. Path
2. List of 11 items
Tree Statistics The Tree Statistics component (Sets/Tree/Tree Statistics) Returns some statistics of the Data Tree
including:
P - All the paths of the tree
L - The length of each branch in the tree
C - Number of paths and branches in the tree
If we connect the Points output of the same Divide Curve component, we can display the paths, lengths, and the
count in panels. This component is helpful because it separates the statistics into three outputs, allowing you to
view only the one that is relevant.
Both the Param Viewer and the Tree Statistics component are helpful for visualizing changes in the structure of the
Data Tree. In the next section, we will look at some operations that can be performed to change this structure.
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Let’s look at some data tree manipulations and visualize how they affect the tree.
Flattening removes all levels of a Data Tree, resulting in a single List. Using the Flatten component
(Sets/Tree/Flatten) on the P output of our Divide Curve component, we can use the Param Viewer to visualize the
new data structure.
In the Param Viewer, we can see that we now only have 1 branch containing a list of 48 points.
Grafting creates a new Branch for every Data Item. If we run the data through the Graft Tree component
(Sets/Tree/Graft Tree), each division point now has its own individual branch, rather than sharing a branch with the
other division points on the same curve.
In the Param Viewer, we can see that what was data with 8 branches of 6 items each, we now have 8
branches with 6 sub-branches containing 1 item each.
Simplify removes overlapping Branches in a Data Tree. If we run the data through the Simplify Tree component
(Sets/Tree/Simplify Tree), the first branch, containing no data, has been removed.
1.5.3. Creating Data Trees
Grasshopper contains tools for changing the structure of a data tree. Theese tools can help you access
specific data within a tree, and change the way it is stored, ordered, and identified.
1.5.3.1. FLATTEN
1.5.3.2. GRAFT TREE
1.5.3.3. SIMPLIFY TREE
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In the Param Viewer, we still have 8 branches of 6 items each, but the first branch has been removed.
The Flip Matrix component (Sets/Tree/Flip Matrix) Swaps the “Rows” and “Columns” of a Data Tree with two Path
Indices.
In the Param Viewer, we can see that what was data with 8 branches of 6 items each, we now have 6
branches with 8 items each.
The Flatten, Graft, and Simplify operations can be applied to the component input or output itself, rather than
feeding the data through a separate component. Just right-click the desired input or output and select Flatten, Graft,
or Simplify from the menu. The component will display an icon to indicate that the tree is being modified. Keep in
mind Grasshopper’s program flow. If you flatten a component input, the data will be flattened before the component
operation is performed. If you flatten a component output, the data will be flattened after the component performs its
action.
1.5.3.4. FLIP MATRIX
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1. Flattened output P
2. Grafted output P
3. Simplified output P
The Path Mapper component (Sets/Tree/Path Mapper) allows you to perform lexical operations on data trees.
Lexical operations are logical mappings between data paths and indices which are defined by textual (lexical)
masks and patterns.
1.5.3.5. THE PATH MAPPER
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1. The Path Mapper component
2. Right-click the Path Mapper component and select a predefined mapping option from the menu, or open
the mapping editor
3. The Mapping Editor
4. You can modify a data tree by re-mapping the path index and the desired branch
Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
In this example, we will manipulate lists and data trees to weave lists of points, define a pattern, and create surface
geometry.
1.5.3.6. WEAVING DEFINITION
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1. Revolved NURBS surface
2. NURBS curve
3. Curve array
4. Division points
5. Paths (indices) of points
1. Array curves
2. Dispatch curves into lists A and B, divide curves
3. Cull points, weave, and revolve
01. Start a new definition, type Ctrl+N (in Grasshopper)
02. Curve/Primitive/Line SDL – Drag and drop the Line SDL component onto the canvas
03. Vector/Point/Construct Point – Drag and drop the Construct Point component ontothe canvas
04. Connect the Point (Pt) output of the Construct Point component to the Start (S) Inputof the Line SDL component
05.
Vector/Vector/Unit Y – Drag and drop the vector Unit Y component onto the canvas
The factor of Unit Vector components is 1.0 by default.
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05. The factor of Unit Vector components is 1.0 by default.
06. Connect the Unit Y component to the Direction (D) input of the Line SDL component
07. Params/Input/Number Slider – Drag and drop the Number Slider component ontothe canvas
08.
Double-click on the Number Slider and set the following:
Name: Length
Rounding: Integer
Lower Limit: 0
Upper Limit: 96
Value: 96
09. Connect the Number Slider to the Length (L) input of the Line SDL component
10. Transform/Array/Linear Array – Drag and drop the Linear Array component ontothe canvas
11. Connect the Line (L) output of the Line SDL component to the Geometry (G) input ofthe Linear Array component
12. Vector/Vector/Unit X – Drag and drop the vector Unit X component onto the canvas
13. Params/Input/Number Slider – Drag and drop two Number Slider components ontothe canvas
14.
Double-click on the first Number Slider and set the following:
Name: Offset Distance
Rounding: Integer
Lower Limit: 1
Upper Limit: 10
Value: 4
15.
Double-click on the second Number Slider and set the following:
Name: # of Offsets
Rounding: Even
Lower Limit: 2
Upper Limit: 20
Value: 20
16. Connect the Number Slider (Offset Distance) to the Factor (F) input of the Unit Xcomponent
17. Connect the Vector (V) output of the Unit X component to the Direction (D) input of theLinear Array component
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18. Connect the Number Slider (# of Offsets) to the Count (N) input of the Linear Arraycomponent
You should now see an array of lines in the Rhino viewport. The three sliders allow you to change the length
of the lines, their distance from each other, and the number of lines in the array.
19. Sets/Lists/Dispatch – Drag and drop the Dispatch component onto the canvas
20. Connect the Geometry (G) output of the Linear Array component to the List (L) inputof the Dispatch component
21. Params/Input/Panel – Drag and drop the Panel component onto the canvas
22.
Double-click the Panel, deselect Multiline Data, Wrap Items and Special Codes, and
enter the following:
true
false
23. Connect the Panel to thePattern (P) input of the Dispatch component
24. Curve/Division/Divide Curve – Drag and drop two Divide Curve components onto thecanvas
25. Connect the List A (A) output of the Dispatch component to the Curve (C) input of thefirst Divide Curve component
26. Connect the List B (B) output of the Dispatch component to the Curve (C) input of thesecond Divide Curve component
27. Params/Input/Number Slider – Drag and drop the Number Slider component ontothe canvas
28.
Double-click on the Number Slider and set the following:
Name: Divisions
Rounding: Integer
Lower Limit: 0
Upper Limit: 20
Value: 20
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29. Connect the Number Slider (Divisions) to the Count (N) input of both Divide Curvecomponents.
1. The Dispatch component sends every second curve in the array to a separate list.
2. The Divide Curve component divides the curves into the number of segments specified by the slider.
Adjust the slider to change the number of points.
30. Sets/Sequence/Cull Pattern – Drag and drop two Cull Pattern components onto thecanvas
31. Connect the Points (P) output of the first Divide Curve component to the List (L) inputof the first Cull Pattern component
32. Connect the Points (P) output of the second Divide Curve component to the List (L)input of the second Cull Pattern component
33. Params/Input/Panel – Drag and drop a second Panel component onto the canvas
34.
Double-click the second Panel and deselect: Multiline Data, Wrap Items, and Special
Codes. Then enter the following:
1
1
0
0
We are using 1 and 0 in place of true and false. These are the two syntaxes
that Grasshopper accepts for boolean values.
35. Connect the second Panel to the Pattern (P) input of the first Cull Pattern component
36. Connect the second Panel to the Pattern (P) input of the second ull Patterncomponent
37.
Right-click on the Pattern (P) input of the second Cull Pattern component and select
Invert 
This will invert the **Cull Pattern**, a useful trick to keep definitions short.
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38. Sets/List/Weave – Drag and drop the Weave component onto the canvas
39. Connect the second Panel to the Pattern (P) input of the Weave component
40. Right-click the Pattern (P) input of the Weave component and select reverse
41. Connect the List (L) output of the first Cull Pattern component to the Stream 0 (0)input of the Weave component
42. Connect the List (L) output of the second Cull Pattern component to the Stream 0 (0)input of the Weave component
43. Curve/Spline/Nurbs Curve – Drag and drop the Nurbs Curve component onto thecanvas
44. Connect the Weave (W) output of the Weave component to the Vertices (V) input ofthe Nurbs Curve component.
1. The cull patterns remove alternating points from each list.
2. The Weave component collects data from the point lists according to a custom pattern. This data is fed
into the interpolate component to create curves.
45. Surface/Freeform/Revolution – Drag and drop two Revolution components onto thecanvas
46. Connect the Curve output of the Nurbs Curve component to the Profile Curve (P)input of both Revolution components.
47. Right Click on Axis (A) input of both Revolution components and select Graft.
48. Connect the List A (A) output of the Dispatch component to the Axis (A) input of thefirst Revolution component
Connect the List B (B) output of the Dispatch component to the Axis (A) input of the
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49.
second Revolution component 
Select all the components except the two Revolution components and turn the
preview off - it is helpful to turn previews off as you build the definition to focus
on the most recent geometry
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
In this example, we will use some of Grasshopper’s tools for manipulating data trees to retreive, reorganize, and
interpolate the desired points contained in a data tree and create a lattice of intersecting fins.
1. Sweep with two rails to create a NURBS surface.
2. Divide the surface into variable sized segments, extract vertices. Data comprised of one list with four
items in each segment.
3. Flip the matrix to change the data structure. Data comprised of four lists, each containing a single corner
point of each segment.
4. Explode the tree to connect corner points and draw diagonal lines across each segement.
5. Prune the tree to cull branches containing insufficient points to construct a degree 3 NURBS curve and
interpolate points.
6. Extrude the curves to create intersecting fins.
01. Start a new definition, type Ctrl+N (in Grasshopper)
02. Params/Geometry/Curve – Drag and drop three curve parameters onto the canvas
03. Surface/Freeform/Sweep2 – Drag a Sweep2 component onto the canvas
Right-click the first Curve parameter and select “Set one curve.” Select the first rail
1.5.4. Working with Data Trees
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curve in the Rhino viewport
05. Right-click the second Curve parameter and select “Set one curve.” Select thesecond rail curve in the Rhino viewport
06. Right-click the third Curve parameter and select “Set one curve.” Select the sectioncurve in the Rhino viewport
07. Connect the outputs of the Curve parameters to the Rail 1 (R1), Rail 2 (R2), andSections (S) inputs of the Sweep2 respectively
We have just created a NURBS surface
08. Params/Geometry/Surface – drag a Surface parameter to the canvas
09. Connect the Brep (S) output of the Sweep2 component to the input of the Surfaceparameter
10.
Right-click the Surface parameter and select “Reparameterize”. 
In this step, we re-mapped the u and v domains of the surface between 0 and
1. This will make future operations possible.
11. Maths/Domain/Divide Domain2 – drag and drop a Divide Domain2 component ontothe canvas
12. Params/Input/Number Slider – drag two Number Sliders onto the canvas
13.
Double click the first Number Sliders and set the following:
Rounding: Integer
Lower Limit: 1
Upper Limit: 40
Value: 20
14. Set the same values on the second Number Sliders
15. Connect the output of the reparameterized Surface parameter to the Domain (I) inputof the Divide Domain2 component
16. Connect the first Number Sliders to the U Count (U) input of the Divide Domain2component
17. Connect the second Number Sliders to the V Count (V) input of the Divide Domain2component
18. Surface/Util/Isotrim – Drag and drop the Isotrim component onto the canvas
19. Connect the Segments (S) output of the Divide Domain2 component to the Domain(D) input of the Isotrim component
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20. Connect the output of the Surface parameter to the Surface (S) input of the Isotrimcomponent
We have now divided out surface into smaller, equally sized, surfaces. Adjust the U and V Count sliders to
change the number of divisions. Lets add a Graph Mapper to give the segments variable size.
21. Maths/Domain/Deconstruct Domain2 – Drag a Deconstruct Domain2 componentonto the canvas
22. Maths/Domain/Construct Domain2 – Drag a Construct Domain2 component to thecanvas
23. Params/Input/Graph Mapper – Drag a Graph Mapper to the canvas
24. Sets/List/List Length – Drag a List Length component to the canvas
25. Sets/Tree/Merge – Drag a Merge component to the canvas
26.
Sets/List/Split List – Drag a Split List component to the canvas 
The Merge and Split components are used here so that the same Graph
Mapper could be used for both the U min and U max values.
27. Connect the U min (U0) and U max (U1) outputs of the Deconstruct Domain2component to the Data 1 (D1) and Data 2 (D2) inputs of the Merge component
28. Connect the Result (R) output of the Merge component to the input of the GraphMapper
29. Right-clickthe Graph Mapper and select “Bezier” under “Graph Types”
30. Connect a second wire from the U max (U1) output of the Deconstruct Domain2component to the List (L) input of the List Length component
31. Connect the Graph Mapper output to the List (L) input of the Split List
32. Connect the Length (L) output of the List Length component to the Index (i) input ofthe Split List component
33. Connect the List A (A) output of the Split List component to the U min (U0) input of theConstruct Domain2 component
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34. Connect the List B (B) output of the Split List component to the U max (U1) input of
the Construct Domain2 component
35. Connect the V min (V0) output of the Deconstruct Domain2 component to the V min(V1) input of the Construct Domain2 component
36. Connect the V max (V1) output of the Deconstruct Domain2 component to the V max(V1) input of the Construct Domain2 component
37. Connect the 2D Domain (I2) output of the Construct Domain2 component to theDomain (D) input of the Isotrim component, replacing the existing connection
We have just deconstructed the domains of each surface segment, remapped the U values using a Graph
Mapper, and reconstructed the domains. Adjust the grips of the Graph Mapper to change the distribution of
the surface segments. Let’s use Data Trees to manipulate the surface divisions.
38. Surface/Analysis/Deconstruct Brep – Drag the Deconstruct Brep component ontothe canvas
39. Sets/Tree/Flip Matrix – Drag the Flip Matrix component to the canvas
40. Sets/Tree/Explode Tree – Drag the Explode Tree component to the canvas
41.
Connect the Surface (S) output of the Isotrim component to the Brep (B) input of the
Deconstruct Brep component 
The Deconstruct Brep component deconstructs a Brep into Faces, Edges, and
Vertices. This is helpful if you want to operate on a specific constituent of the
surface.
42.
Connect the Vertices (V) output of the Deconstruct Brep component to the Data (D)
input of the Flip Matrix component 
We just changed the Data tree structure from one list of four vertices that
define each surface, to four lists, each containing one vertex of each surface.
43. Connect the Data (D) output of the Flip Matrix component to the Data (D) input of theExplode Tree component
44. Right-click the Explode Tree component and select “Match Outputs”
45. Right-click the Data (D) input of the Explode Tree component and select simplify
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Each output of the Explode Tree component contains a list of one vertex of each surface. In other words, one
list with all the top right corners, one list with all the bottom right corners, one list of top left corners, and one
list of bottom left corners.
46. Curve/Primitive/Line – Drag and drop two Line components onto the canvas
47. Connect the Branch 0 {0} output of the Explode Tree component to the Start Point (A)input of the first Line component
48. Connect the Branch 1 {1} output of the Explode Tree component to the Start Point (A)input of the second Line component
49. Connect the Branch 2 {2} output of the Explode Tree component to the End Point (B)input of the first Line component
50. Connect the Branch 3 {3} output of the Explode Tree component to the End Point (B)input of the second Line component
We have now connected the corner points of each surface diagonally with lines.
51. Curve/Util/Join Curves – Drag and drop the Join Curves component to the canvas
52. Curve/Analysis/Control Points – Drag a Control Points component onto the canvas
53. Curve/Spline/Interpolate – Drag and drop the Interpolate component onto thecanvas
54.
Connect the Line (L) outputs of each Line component to the Curves (C) input of the
Join Curves component 
Hold down the Shift key to connect multiple wires to a single input
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55. Connect the Curves (C) output of the Join Curves component to the Curve (C) input
of the Control Points component
56. Connect the Points (P) output of the Control Points component to the Vertices (V)input of the Interpolate component
We have now joined our lines into polylines and reconstructed them as NURBS curves by interpolating their
control points. In the Rhino viewport, you might notice that the shorter curves are still straight lines. This is
because you cannot make a degree three NURBS curve with fewer than four control points. Let’s manipulate
the data tree to eliminate lists of control points with less than four items.
57. Sets/Tree/Prune Tree – Drag and drop the Prune Tree component onto the canvas
58. Params/Input/Panel – Drag a Panel onto the canvas
59.
Connect the Points (P) output of the Control Points component to the Tree (T) input of
the Prune Tree component 
If you connect one Param Viewer to the Points (P) output of the Control Points
component, and another to the Tree (T) output of the Prune Tree component,
you can see that the number of branches has been reduced.
60. Double click the Panel and enter 4.
61. Connect the output of the Panel to the Minimum (N0) input of the Prune Treecomponent
62. Connect the Tree (T) output of the Prune Tree component to the Vertices (V) input ofthe Interpolate component
63. Surface/Freeform/Extrude – Drag and drop the Extrude component onto the canvas
64.
Vector/Vector/Unit Y – Drag a Unit Y component onto the canvas
You may need to use a Unit X vector, depending on the orientation of your referenced
geometry in Rhino
65. Params/Input/Number Slider – Drag a Number Slider onto the canvas
66.
Double click the Number Slider and set the following:
Rounding: Integer
Lower Limit: 1
Upper Limit: 5
Value: 3
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67. Connect the Curve (C) output of the Interpolate component to the Base (B) input ofthe Extrude component
68. Connect the Number Slider output to the Factor (F) input of the Unit Y component
69. Connect the Unit Vector (V) output of the Unit Y component to the Direction (D) inputof the Extrude component
You should now see a diagonal grid of strips or fins in the Rhino Viewport. Adjust the Factor slider to chnage
the depth of the fins
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1.6. Getting Started with Meshes
In the field of computational modeling, meshes are one of the most pervasive
forms of representing 3D geometry. Mesh geometry can be a light-weight and
flexible alternative to working with NURBS, and are used in everything from
rendering and visualizations to digital fabrication and 3D printing. This chapter
will provide an introduction to how mesh geometry is handled in Grasshopper.
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145Getting Started with Meshes
1. Mesh vertices
2. Mesh edges
3. Mesh faces
Grasshopper defines meshes using a Face-Vertex data structure. At its most basic, this structure is simply a
collection of points which are grouped into polygons. The points of a mesh are called vertices, while the polygons
are called faces. To create a mesh we need a list of vertices and a system of grouping those vertices into faces.
1. A list of vertices.
2. Faces with groupings of vertices
1.6.1 What is a Mesh?
A Mesh is a collection of quadrilaterals and triangles that represents a surface or solid geometry. This section
discusses the structure of a mesh object, which includes vertices, edges, and faces, as well as additional
mesh properties such as colors and normals.
1.6.1.1 Basic Anatomy of a Mesh
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Vertices
The vertices of a mesh are simply a list of points. Recall that a list in Grasshopper is a collection of objects. Each
object in the list has an index which describes that objects position in a list. The index of the vertices is very
important when constructing a mesh, or getting information about the structureof a mesh.
1. A list of points. All lists in Grasshopper begin with an index of zero
2. The set of points labeled with their index
Faces
A face is an ordered list of three or four vertices. The “surface” representation of a mesh face is therefore implied
according to the position of the vertices being indexed. We already have the list of vertices that make up the mesh,
so instead of providing individual points to define a face, we instead simply use the index of the vertices. This also
allows us to use the same vertex in more than one face.
1. A quad face made with indices 0, 1, 2, and 3
2. A triangle face made with indices 1, 4, and 2
In Grasshopper, faces can be created with the Mesh Triangle and Mesh Quad components. The input for these
components are integers that correspond to the index of the vertices we want to use for a face. By connecting a
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147What is a Mesh?
Panel to the output of these components, we can see that a triangular face is represented as T{A;B;C}, and a quad
face as Q{A;B;C;D}. Faces with more than 4 sides are not allowed. To make a 5-sided mesh element, the mesh
must be broken into two or more faces.
1. Mesh Quad component with indices 0, 1, 2, and 3
2. Mesh Triangle component with indices 1, 4, and 2
It is important to remember that these components do not result in the creation of mesh geometry, rather the output
is a list of indices that define how a mesh should be constructed. By paying attention to the format of this list, we
can also create a face manually by editing a Panel component and entering the appropriate format for either
triangular or quad faces.
1. A face created using a Mesh Quad component
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2. A face created using a Panel
3. A Panel Properties window is automatically opened when double-clicking a panel while zoomed out, or
by right-clicking a panel and selecting "Edit Notes..."
So far we have a list of vertices and a set of face definitions, but have not yet created a mesh. In order to create a
mesh, we need to connect the faces and vertices together by using the Construct Mesh component. We connect
our list of vertices to the V input, and a merged list of faces to the F input. (The component also has room for an
optional Color input, which is discussed below.) If we connect a panel to the output of the Construct Mesh we can
see information about the number of faces and number of indices.
1. The Construct Mesh component takes a list of vertices and a list of faces as input. The Color input is
optional, and is left blank for now
2. A panel shows that we have created a mesh with 5 vertices and 2 faces
3. The resulting mesh (the vertices have been labeled with their indices)
By default, Grasshopper does not preview the edges of mesh geometry. To preview the edges as well as the
surfaces, you can turn on mesh edge preview by using the shortcut Ctrl-M, or by going to the Display menu and
selecting 'Preview Mesh Edges'.
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It is extremely important to pay attention to the order of the indices when constructing a mesh face. The face will be
constructed by connecting the vertices listed in order, so the quad faces Q{0,1,2,3} and Q{1,0,2,3} are very different,
despite using the same four vertices. Incorrect vertex ordering can lead to problems such as holes, non-manifold
mesh geometry, or non-orientable surfaces. Such mesh geometry is usually not correctly rendered, and not able to
be 3D printed. These issues are discussed in more detail in the Understanding Topology section.
1. A quad face with indices 0,1,2,3
2. A quad with indicies 0,3,1,2
1.6.1.2 Implicit Mesh Data
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In addition to faces and vertices, there is other information about a mesh that we will want to use. In a Face-Vertex
based mesh, data such as edges and normals are calculated implicitly based on the given faces and vertices. This
section describes ways to query this information.
Edges
The edges of a mesh are lines connecting any two consecutive vertices in a face. Notice that some edges are
shared between multiple faces, while other edges are only adjacent to one face. The number of faces an edge is
adjacent to is called the valence of that edge.
Grasshopper groups edges into three categories based on the valence:
1. E1 - 'Naked Edges' have a valence of 1. They make up the external boundary of a mesh.
2. E2 - 'Interior Edges' have a valence of 2.
3. E3 - 'Non-Manifold Edges' have a valence of 3 or greater. Meshes that contain such structure are called "Non-
Manifold", and are discussed in the next section.
1. Naked edge with valence of 1
2. Interior edge with valence of 2
3. Non-manifold edge with valence of 3
We can use the Mesh Edges component to get the edges of a mesh outputted according to valence. This allows us
to locate edges along the boundary of a mesh, or to identify non-manifold edges. Sometimes, however, it is more
useful to have the full boundary of each face. For this, we can use the Face Boundaries component. This will return
a polyline for each face.
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1. The Mesh Edges component outputs three sets of edges. This mesh has 5 naked edges, 1 interior
edge, and zero non-manifold edges
2. The E3 output is empty, because this mesh does not have any non-manifold edges, resulting in an
orange wire.
3. The Face Boundaries component outputs one polyline for each face
Face Normals
A normal vector is a vector with a magnitude of one that is perpendicular to a surface. In the case of triangular faces,
we know that any three points must be planar, so the normal will be perpendicular to that plane, but how do we
know which direction ('up' or 'down') the normal will be pointing? Once again, the order of the indicies is crucial
here. Mesh faces in Grasshopper are defined counter-clockwise, so a face with indices {0,1,2} will be 'flipped' as
compared to the indicies {1,0,2}. Another way to visualize this is to use the Right-Hand-Rule.
1. The Face Normals component will return a list of center points and normal vectors for each face
2. Face normals according to vertex sequence
3. "Right-Hand-Rule" for determining normal direction
Grasshopper also allows quad faces, in which case the 4 points will not always be planar. For these faces, the
center point will be simply the average of the coordinates of the 4 vertices (in the case of a non-planar quad, note
that this point is not necessarily on the mesh). To calculate the normal of a quad face, we need to first trianglulate
the quad by splitting it into two planar triangles. The normal of the overall face is then the average of the two
normals, weighted according to the area of the two triangles.
Vertex Normals
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In addition to the face normals, it is also possible to calculate normals for each vertex of a mesh. For a vertex that is
only used in a single face, the normal at the vertex will point in the same direction as the face normal. If a vertex has
multiple adjacent faces, the vertex normal is calculated by taking the average of the faces.
While less intuitive than face normals, vertex normals are important for smooth visualization of meshes. You might
notice that even when mesh is composed of planar faces, such a mesh can still appear smooth and rounded when
shaded in Rhino. Using the vertex normals allows this smooth visualization.
1. Normals set according to the face normal results in discrete polygonal shading
2. Adjancent face normals are averaged together to create vertex normals, resulting in smooth shading
across faces
Meshes can also be assigned additional attributes to either vertices or faces. The simplest of these is vertex color,
which is described below, but other attributes exist such as texture UV coordinates. (Some programs even allow
vertex normals to be assigned as attributes instead of being derived from the faces andvertices, which can provide
even more flexibility in rendered surface appearance.)
Color
When using a Construct Mesh component, there is an option input for vertex color. Colors can also be assigned to
an existing mesh using the Mesh Color component. By using a single color for a mesh, we can color the entire
mesh.
Trianglular mesh objects colored with red, green, or blue
While the above examples colored the entire mesh, color data are actually assigned for each vertex. By using a list
1.6.1.3 Mesh Attributes
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153What is a Mesh?
of three colors, we can color each vertex in the triangle separately. These colors are used for visualitizations, with
each face rendered as an interpolation of the vertex colors. For example, the image below shows a triangular face
with vertex colors of Red, Green, and Blue.
1. Red, green, and blue are assigned to the three vertices of a mesh
2. The resulting mesh interpolates the colors of the vertices
Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
01. Start a new definition, type Ctrl-N (in Grasshopper)
02. Mesh/Primitive/Mesh Quad - Drag and drop a Mesh Quad component onto the canvas
03. Mesh/Primitive/Construct Mesh - Drag and drop a Construct Mesh component onto thecanvas
04. Connect the Face (F) output of the Mesh Quad component to the Faces (F) input of theConstruct Mesh component
Mesh Quad and Construct Mesh have default values which create a single mesh face. Next, we will replace
the default values with our own vertices and faces.
1.6.1.4 Exercise
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http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
05. Params/Input/Panel - Drag and drop a Panel component onto the canvas
06. Double-click the Panel component and set the value to '0'
07.
Params/Input/Panel - Drag and drop four more Panel components onto the canvas and
set their values to 1,2,3, and 4 
You can also copy the original **Panel** by clicking and dragging, then tapping the
Alt key before releasing the click
08.
Connect the Panels to the inputs of the Mesh Quad in the following order:
0 - A
1 - B
2 - C
3 - D
09. Mesh/Primitive/Mesh Triangle - Drag and drop a Mesh Triangle component onto thecanvas
10.
Connect the Panels to the inputs of the Mesh Triangle component in the following order:
1 - A
2 - B
4 - C
11. Sets/Tree/Merge - Drag and drop a Merge component onto the canvas
12.
Connect the Face (F) output of the Mesh Quad component to the Data1 (D1) input of the
Merge component, and the Face (F) output of the Mesh Triangle component to the Data2
(D2) input of the Merge component
13. Connect the Result (R) output of the Merge component to the Faces (F) input of theConstruct Mesh component
The default Vertices (V) list of Construct Mesh only has 4 points, but our Mesh Triangle component uses an
index of 4, which would correspond to the fifth point in a list. Since there are not enough vertices, the
Construct Mesh component gives an error. To fix it, we will provide our own list of points.
14. Params/Input/Panel - Drag and drop a Panel component onto the canvas
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15.
Right-click the Panel component and de-select the 'Multiline Data' option
By default, a panel has 'Multiline Data' enabled. By disabling it, each line in the panel
will be read as a separate item within a list.
16.
Double-click the Panel component to edit it, and enter the following points:
{0,0,0}
{1,0,0}
{1,1,0}
{0,1,0}
{2,0,0}
Make sure you use the correct notation. To define a point in a **Panel**, you have to
use curly brackets: '{' and '}' with commas between the x, y, and z values
17. Connect the Panel component to the Vertices (V) input of the Construct Mesh component
We now have a mesh with two faces and 5 vertices.
Optionally, we can replace the Mesh Quad and Mesh Triangle components with a panel specifying the indices of
the faces.
18. Params/Input/Panel - Drag and drop a Panel component onto the canvas
19.
Right-click the Panel component and deselect 'Multiline Data' 
Alternatively, copy the existing **Panel** that we used for the points, which already
has 'Multiline Data' disabled
20.
Double-click the Panel component to edit it, and enter the following:
Q{0,1,2,3}
T{1,2,4}
21. Connect the Panel to the Faces (F) input of the Construct Mesh component
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22. Params/Input/Colour Swatch - Drag and drop a Colour Swatch component onto thecanvas
23. Click the colored section of the component (the default is White) to open the color selectionpanel
24. Use the sliders to set the G and B values to zero. The swatch should now be Red
25. Params/Input/Colour Swatch - Drag and drop two more Colour Swatch components ontothe canvas and set their colors to Blue and Green
26. Sets/Tree/Merge - Drag and drop a Merge component onto the canvas
27. Connect the three Color Swatch components into the D1, D2, and D3 inputs of the Mergecomponent.
28. Connect the Result (R) output of the Merge component to the Colours (C) input of theConstruct Mesh component
We have 5 vertices, but only 3 colors. Grasshopper will assign the colors in a repeating pattern, so in this
cases vertices 0 and 3 will be Red, vertices 1 and 4 will be Green, and the final vertex 2 will be Blue.
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Any introduction to mesh geometry would be incomplete without at least a basic introduction to topology. Because
topology is concerned with the inter-relationships and properties of a set of “things” rather than the “things”
themselves, it is mobilized for an enormous range of both tangible and intangible applications. In this primer, we
are interested in its basic application with respect to parametric workflows that afford us the opportunity to create
and control mesh geometry.
In Grasshopper, the two basic types of information required to define a mesh are geometry and connectivity; in other
words, a set of points in rhino-space (vertices) and set of corresponding point-associations (faces).
Without connectivity information, a mesh is unstructured and therefore still undefined. The introduction of a set of
faces is the step (or leap) that ultimately actualizes a mesh and establishes its character in terms of continuity,
convergence, and connectedness; this structural network is referred to as a topological space.
The same set of vertices can have different connectivity information, resulting in different topology.
1.6.2 Understanding Topology
While the vertices of a mesh contain position information, it is really the connections between the vertices that
give a mesh geometry its unique structure and flexibility.
1.6.2.1 What is Topology?
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Homeomorphism
The points of a mesh can be moved without altering the connectivity information. The new mesh has the
same topology as the original.
It is possible for two distinct mesh shapes to be topologically identical. All this would mean is that they are
constructed out of the same number of points and that the points are structured by the same set of faces. Earlier,
we established that a mesh face is only concerned with the indices of a set of points and has no interest in their
actual location in rhino-space. Therefore, if the only difference between two distinct mesh shapes is the specific 3-
dimentional position of the points that are used to define it, then the two meshes are considered to be
“homeomorphic” (or topologically equivalent) and therefore share the same topological properties.
An example of homeomorphism among letters (note that some of the above homeomorphic groups might be
different depending on what font is considered)
A topologically equivalent mug and donut
1.6.2.2 Mesh Characteristics
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Orientable
A mesh is considered orientab le if there are well-defined sides to themesh. An simple example of a non-orientable
mesh occurs when adjacent faces have normals pointing in opposite directions. These 'flipped faces' can cause
problems in visualizations and renderings, as well as manufacturing or 3D-printing.
1. An orientable surface with face normals pointing in the same direction.
2. A non-orientable surface has adjacent normals pointing in different directions.
Open vs Closed
It is often necessary to know whether a mesh is a closed mesh which represents a volumetric solid, or an open
mesh that just represents a 2-dimensional surface. The difference can be imperative with respect to
manufacturability. You cannot 3D print a single surface which has no thickness, but must instead thicken the mesh
so that it is a solid. Solid mesh geometry is also required for Boolean operations (discussed in following section).
The Mesh Edges component can be used to help determine this. If none of the edges of a mesh have a valence of
1 (if the E1 output is null), then we know that all the edges are 'Interior Edges' and the mesh does not have an
external boundary edge, and is therefore a closed mesh.
On the other hand, if there exist 'Naked Edges', then those edges must be on a boundary of the mesh, and the
mesh is not closed.
1. A closed mesh. All edges are adjacent to exactly two faces.
2. An open mesh. The white edges are each adjacent to only a single face.
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Manifold vs Non-Manifold
Non-manifold geometry is essentially geometry that cannot exist in the "real world". This does not necessarily make
it "bad geometry" but it is something to be aware of due to complications it may present for tools and operations (for
example: rendering of refractive effects, fluid simulations, boolean operations, 3d printing, etc). Common conditions
that result in a non-manifold mesh include: self intersection, naked edges (from holes or internal faces), disjoint
topology, and overlapping/duplicate faces. A mesh can also be considered Non-Manifold if it includes any vertices
which are shared by faces that do not share edges or any edges with a valence greater than 2, creating a junction of
at least 3 faces
1. A simple manifold mesh
2. Three faces meeting on a single edge is non-manifold, also known as a T-Junction
3. Two faces meeting at only one vertex but not sharing an edge is non-manifold
How is mesh geometry different from NURBS geometry? When might you want to use one instead of the other?
In a previous chapter, we saw that NURBS surfaces are defined by a series of NURBS curves going in two
directions. These directions are labeled U and V, and allow a NURBs surface to be parameterized according to a
two-dimensional surface domain. The curves themselves are stored as equations in the computer, allowing the
resulting surfaces to be calculated to an arbitarily small degree of precision. It can be difficult, however, to combine
multiple NURBS surfaces together. Joining two NURBS surfaces will result in a polysurface, where different
sections of the geometry will have different UV parameters and curve definitions.
Meshes, on the other hand, are comprised of a discrete number of exactly defined vertices and faces. The network
of vertices generally cannot be defined by simple UV coordinates, and because the faces are discrete the amount
of precision is built into the mesh and can only be changed by refining the mesh and added more faces. The lack of
UV coordinates, however, allows meshes the flexibility to handle more complicated geometry with a single mesh,
instead of resorting to a polysurface in the case of NURBS.
Note - While a mesh does not have implicit UV parameterization, it is sometimes useful to assign such a
parameterization in order to map a texture or image file onto mesh geometry for rendering. Some modeling
software therefore treats the UV coordinates of a mesh vertex as an attribute (like vertex color) which can be
manipulated and changed. These are usually assigned and not completely defined by the mesh itself.
Another important difference is the extent to which a local change in mesh or NURBS geometry affects the entire
form. Mesh geometry is completely local. Moving one vertex affects only the faces that are adjacent to that vertex. In
NURBS surfaces, the extent of the influence is more complicated and depends on the degree of the surface as well
as the weights and knots of the control points. In general, however, moving a single control point in a NURBS
surface creates a more global change in geometry.
1.6.2.3 Meshes Vs NURBS
Parameterization
Local vs Global Influence
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1. NURBS Surface - moving a control point has global influence
2. Mesh geometry - moving a vertex has local influence
One analogy that can be helpful is to compare a vector image (composed of lines and curves) with a raster image
(composed of individual pixels). If you zoom into a vector image, the curves remain crisp and clear, while zooming
into a raster image results in seeing individual pixels. In this analogy, NURBS surfaces can be compared to a
vector image, while a mesh behaves similarly to a raster image.
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Zooming into a NURBS surface retains a smooth curve, while a mesh element has a fixed resolution
It is interesting to note that while NURBs surfaces are stored as mathematical equations, the actual visualization of
these surfaces requires meshes. It is not possible for a computer to display a continuous equation. Instead, it must
break that equation down into smaller parts, the result of which is that all rendering or display processing must
convert NURBS to meshes. As an analogy, consider that even though we can store the equation of a line on a
computer, in order to display that line, the computer must at some point convert the line into a series of discrete
pixels on a screen to display.
When we ask “What are the pros and cons of modeling with meshes?” we are really asking “What are the pros and
cons of modeling with shapes that are defined solely by a set of vertices and a corresponding topological
framework?” Through this method of framing the question it is easier to see how the “simplistic” nature of a mesh
is the critical aspect that would make a mesh either favorable or unfavorable to model with depending on the
context of its application.
Meshes can be favorable in situations where:
There must be a dynamically updated rendering of a form that is changing in shape but not in face connectivity
A discretized approximation of a curved geometry would suffice
A low-resolution geometry must be systematically smoothed (or articlated) using computational methods to
arrive at a higher-resolution model.
The low resolution model must be able to be to simultaneously support local, high resolution detail
Meshes can be unfavorable in situations where:
Curvature and smoothness must be represented with a high level of precision
True derivatives must be evaluated
The geometry must be converted into a manufacturable solid
The final form must be able to be easily edited manually
1.6.2.4 Pros and Cons of Meshes
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There are three fundamental ways of creating mesh geometry in Grasshopper:
1. Starting with a mesh primitive
2. Manually constructing a mesh from faces and vertices
3. Converting NURBS geometry into a mesh
Grasshopper comes with a few simple mesh primitive components:
1. Mesh Box - This primitve requires a Box object as an input which provides the size and location, as well
as X,Y, and Z values that determine how many faces to divide the box into. The six sides of a Mesh Box
are unwelded allowing for creases. (See the following section for more information about welded
meshes)
2. Mesh Plane - This primitive requires a Rectangle input to determine the size and location of the plane,
as well as W and H values to determine the number of faces.
3. Mesh Sphere- This primitive requires a base plane to determine the center and orientation of the
sphere, a radius for the size, and U and V values to determine the number of faces.
4. Mesh Sphere Ex - Also known as a Quadball, this primitive creates a sphere composed of six patches,
which are subdivided according to the C input. A quadball is topologically equivalent to a cube, even
though it is geometrically spherical.
As we saw in the previous section, the Construct Mesh component can be used to directly create a mesh from a
list of vertices and a list of faces (and an optional list of vertex colors). Constructing an entire mesh manually can be
extremely tedious, so this component is more often used with an existing list of faces and vertices which have been
1.6.3 Creating Meshes
In the last section, we looked at the basic structure of meshes. In this section, we give a brief introduction to
different ways of generating mesh geometry.
1.6.3.1 Primitive
1.6.3.2 Construct Mesh
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extracted using a Deconstruct Mesh component on an existing mesh.
Perhaps the most common method of creating a complex mesh is to generate one based off of NURBS geometry.
Individual NURBS surfaces can be converted to a mesh using the Mesh Surface component, which simply
subdivides the surface along its UV coordinates and creates quad faces. This component allows you to enter the
number of U and V divisions for the resultant mesh.
More complex polysurfaces can be converted to a single mesh with the Mesh Brep component. This component
also has an optional Settings input, which can be set by using one of the built in Speed, Quality, or Custom Settings
components, or by right-clicking the S input and selecting "Set Mesh Options". For efficient use of meshes, it is
often necessary to refine this mesh by using various strategies such as rebuilding, smoothing, or subdividing.
Some of these techniques will be discussed later in this Primer.
1. Mesh Surface converts a NURBS surface to a mesh
2. Mesh Brep can convert polysurfaces and more complicated geometry into a single mesh. Adjusting the
settings will allow for more or less faces, and a finer or coarser mesh.
NOTE: it is generally much easier to convert from a NURBS geometry to a mesh object rather than the other way
around. While the UV coordinates of a NURBS surface are straightforward to convert to quad faces of a mesh, the
opposite is not necessarily true, since a mesh might contain a combination of triangles and quads in a way that is
not simple to extract a UV coordinate system out of.
In this exercise, we use a basic Mesh primitive, perform a transformation on the vertices, and then assign a color
based on the normal vectors to approximate the rendering process.
Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
01. Start a new definition, type Ctrl-N (in Grasshopper)
02. Mesh/Primitive/Mesh Sphere - Drag and drop a Mesh Sphere component onto the canvas
03.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Rounding: Integer
Lower Limit:0
1.6.3.3 NURBS to Mesh
1.6.3.4 Exercise
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http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Upper Limit: 100
Value: 10
04. Connect the Number Slider to the U Count (U) and V Count (V) inputs of the Mesh SphereComponent
Adjust the slider and notice how the resoultion of the sphere changes in the Rhino viewport. Higher numbers
result in a smoother sphere, but also produce larger datasets which can require more processing time.
05. Mesh/Analysis/Deconstruct Mesh - Drag and drop a Deconstruct Mesh component ontothe canvas
06. Connect the Mesh (M) output of the Mesh Sphere component to the Mesh (M) input of theDeconstruct Mesh component
07. Transform/Euclidean/Move - Drag and drop a Move component onto the canvas
08. Connect the Vertices (V) output of the Deconstruct Mesh component to the Geometry (G)input of the Move component
09. Connect the Normals (N) output of the Deconstruct Mesh component to the Motion (T)input of the Move component
10. Mesh/Analysis/Construct Mesh - Drag and drop a Construct Mesh component onto thecanvas
11. Connect the Geometry (G) output of the Move component to the Vertices (V) input of theConstruct Mesh component
12. Connect the Faces (F) output of the Deconstruct Mesh component to the Faces (F) of theConstruct Mesh component
We deconstructed the mesh to get its vertices, faces, and normals. We then simply moved each vertex
according to its normal vector. Because we did not change the topology of the sphere at all, we re-used the
list of faces to re-construct the new mesh. Normal vectors always have a length of one, so this ended up
reconstructing a new mesh sphere with a radius of one more than the original sphere.
Next, we will use a sine function to manipulate the sphere in a slightly more complicated way.
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13. Vector/Point/Deconstruct - Drag and drop a Deconstruct component onto the canvas
14. Connect the Vertices (V) output of the Deconstruct Mesh component to the Point (P) inputof the Deconstruct component
15. Params/Input/Number Slider - Drag and drop two Number Slider components onto thecanvas
16.
Set the values of the frist Number Slider to:
Name: Amplitude
Rounding: Float
Lower Limit: 0
Upper Limit: 10
17.
Set the values of the second Number Slider to:
Name: Frequency
Rounding: Float
Lower Limit: 0
Upper Limit: 5
18. Maths/Script/Expression - Drag and drop an Expression component onto the canvas
19. Zoom in to the Expression component until you see the options for adding or removinginput variables and click on a '+' to add a 'z' variable
20. Right click the 'y' input of the Expression component and change the text to 'A'
21. Right click the 'z' input of the Expression component and change the text to 'f'
22.
Double click the Expression component to edit the expression, and enter the following:
A\*sin(x\*f/π)
23. Connect the X output of the Deconstruct component to the 'x' input of the Expressioncomponent
24. Connect the Amplitude Number Slider to the A input, and the Frequency Number Slider tothe 'f' input of the Expression component
25. Maths/Operators/Multiplication - Drag and drop a Multiplication component onto thecanvas
26. Connect the Normals (N) output of the Deconstruct Mesh component to the A input of theMultiplication component
27. Connect the Result (R) output of the Expression component to the the B input of theMultiplication component
28. Connect the Result (R) output of the Multiplication component to the Motion (T) input of theMove component
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Adjust the Amplitude and Frequency number sliders to see how the newly constructed mesh changes.
29. Mesh/Primitive/Mesh Colours - Drag and drop a Mesh Colours component onto thecanvas
30.
Params/Input/Gradient - Drag and drop a Gradient component onto the canvas 
You can right-click the gradient component and select "Presets" to change the color
gradient. In this example, we used the Red-Yellow-Blue gradient
31. Connect the Result (R) output of the Expression component to the Parameter (t) input ofthe Gradient component
32. Connect the output of the Gradient component to the Colours (C) input of the MeshColours component
33.
Connect the Mesh (M) output of the Construct Mesh component to the Mesh (M) input of
the Mesh Colours component 
In this step, we could achieve the same result by connecting the gradient directly to
the Colours (C) input of the **Construct Mesh** component
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We used the Expression results to drive both the movement of the vertices and the color of the mesh, so the
color gradient in this case corresponds to the magnitude of the movement of the vertices.
For the final portion of the exercise, we will instead usethe direction of the normals relative to a 'light source' vector
to simulate the basic process of rendering a mesh.
34. Mesh/Analysis/Deconstruct Mesh - Drag and drop a Deconstruct Mesh component ontothe canvas
35.
Connect the Mesh (M) output of the Construct Mesh component to the Mesh (M) input of
the Deconstruct Mesh component 
While the topology of the original mesh has not changed, the normal vectors will be
different, so we need to use a new Deconstruct Mesh to find the new normals.
</blockquote
36.
Vector/Vector/Unit Z - Drag and drop a Unit X component onto the canvas 
We will use this as the direction of a light source. You can use other vectors, or
reference a line from Rhino to make this more dynamic
37. Vector/Vector/Angle - Drag and drop an Angle component onto the canvas
38. Connect the Normals (N) output of the Deconstruct Mesh component to the A input of theAngle component
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39. Connect the output of the Unit Z component to the B input of the Angle component
40. Maths/Util/Pi - Drag and drop a Pi component onto the canvas
41. Connect the Pi component to the Upper Limit (L1) input of the Gradient component
42. Connect the Angle (A) output of the Angle component to the Parameter (t) input of theGradient component
We used the white-to-black preset for our gradient. This sets the mesh color according to the angle between
the normal and the light source, with normals that are directly facing the light source to black and the normals
facing away from the source to white (To be a little more accurate, you can reverse the gradient by adjusting
the handles). The actual process of rendering a mesh is much more complicated than this, obviously, but
this is the basic process of creating light and shadow on a rendered object.
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Smoother meshes can sometimes be achieved by simply increasing the number of faces in a process called
subdivision. This can often lead to extremely large datasets which take a long time to calculate, and requires
additional add-ons to Grasshopper that are not built-in. In these situations, the Smooth component can be used as
an alternative to make meshes less jagged or faceted, without increasing vertex and face count or changing the
topology. The strength, number of iterations, and displacement limit can all be used to adjust how much smoothing
occurs.
Attaching a boolean value to the input N provides option to skip naked vertices. A vertex is naked if it is connected to
a naked edge, meaning the vertex is on the boundary of an open mesh. By toggling this option, you maintain the
exterior boundary of a mesh while smoothing the interior edges.
1. Initial box mesh with 3 faces removed
2. Smoothing after 2 iterations
3. 6 iterations
4. 25 iterations
5. 50 iterations
The Blur component acts in a similar way as smooth, except it only affect the vertex colors. It can also be used to
reduce the jagged appearance of colored meshes, although to a lesser extent since it does not change any
1.6.4 Mesh Operations
In the last section, we looked at the basic structure of a mesh. In this section, we will look at ways to
manipulate mesh geometry.
1.6.4.1 Smooth
1.6.4.2 Blur
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geometry.
1. Initial mesh
2. Blur after 1 iterations
3. 6 iterations
4. 12 iterations
5. 20 iterations
In order to ensure each face is planar, or to export a mesh to a different software that might not allow quad faces, it
is sometimes necessary to triangulate a mesh. Using the Triangulate component, each quad face is replaced with
two triangle faces. Grasshopper always uses the shortest diagonal of the face to create a new edge.
1. Original quad mesh
2. Added edges according to shortest distance across quads
3. Triangulated resultant mesh
1.6.4.3 Triangulate
1.6.4.4 Weld
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In the last section, we noticed that a single vertex can be shared by adjacent faces and the normal for that vertex is
calculated as the average of the adjacent faces, allowing a smoother visualization. However, it is sometimes
desireable to have a sharp crease or seam where one face does not smoothly transition to the next by way of the
vertex normals. For this situation it is necessary for each face to have its own vertex with its own normal. The list of
vertices would contain at least two points that have the same coordinates, but different indices.
1. Welded Faces - Both faces share vertices 1 and 2. The vertex normals at these vertices are the average
of the face vertices.
2. Unwelded Faces - Duplicate vertices are added to the list. The faces do not share any vertex indices.
Vertices 1 and 6, and vertices 2 and 5 have identical coordinates, but are separate vertices. They each
have their own vertex normal
The process of taking two vertices that are in the same position and combining them into a single vertex is called
welding, while unwelding takes a single vertex and splits it into multiple vertices.
The Weld component uses a threshold angle as input. Any two adjacent faces with an angle less than the
threshold angle will be welded together, resulting in common vertices with a normal that is the average of the
adjacent faces. Unweld works in the opposite manner, where adjancent faces with an angle greater than the given
threshold will be unwelded, and their shared vertices will be duplicated.
1. The default Box Mesh has 726 vertices. The mesh is creased at the corners of the box, where vertices
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are doubled up.
2. If the mesh is welded with an angle greater than 90 degrees, the resulting mesh faces are welded
together, and the number of vertices has decreased to 602 while the number of faces has remained the
same.
3. Looking at the previewed geometry, you can also notice that the rendered welded mesh has smoothed
corners.
4. Unlike the Smooth component which changes the mesh geometry, this mesh only appears smoother
due to the vertex normal's role in rendering and shading. The actual positions of the vertices remain
unchanged.
In the above image, we used the angle 91 degrees because we know that the sides of a square will be at 90
degree angles. To completely weld an entire mesh you should use an angle of 180 degrees.
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Inclusion
This component tests to determine whether a given point is inside a mesh solid or not. This only works with closed
meshes.
Mesh Closest Point
This component will calculate the position on a mesh that is closest to a given point. This component outputs three
pieces of data: the coordinates of the calculated point on the mesh, the index of the face which contains that point,
and the mesh parameter. This parameter is extremely useful in conjunction with the Mesh Eval component
discussed below.
1. Given a point in space, We want to find the closet point on the mesh
2. The face that contains the closest point is identified
3. The parameters of the closest point on the face are calculated
For those users interested in a little more detail about how a mesh is parameterized, we can take a closer look at
how a mesh parameter is structured. You can see this structure by attaching a panel to the parameter output of a
Mesh Closest Point component. The mesh parameter has the form: N[A,B,C,D]. The first number, N, is the index of
the face which contains the calculated point.
The following four numbers define the barycentric coordinates of the point within that face. The coordinates of the
1.6.5 Mesh Interactions
This section looks at ways in which Mesh Objects can interact with other objects, such as evaluating nearest
points or combining multiple meshes together.
1.6.5.1 Mesh Geometry and Points
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referenced point can be found by multiplying each vertex of the face by these numbers in order and then add the
results together. (Of course,this is already done for us, and is given in the Point output). Also note that barycentric
coordinates are only unique for triangular faces, meaning that on a quad face the same point could have multiple
different parameterizations. Grasshopper avoids this problem by internally triangulating a quad face when
calculating a parameter, the result of which is that of the four numbers in a mesh parameter, at least one of them
will always be zero.
Barycentric Coordinates
Mesh Eval
The Mesh Eval component uses a mesh parameter as an input and returns the referenced point, as well as the
normal and color at that point. The color and normal are calculated as interpolations of the vertex colors and vertex
normals, using the same barycentric coordinates as the mesh parameter.
Mesh Join
Unlike joining curves or NURBS surfaces which require adjacency, any meshes can be joined into a single mesh -
even meshes that are not touching. Recall that a mesh is simply a list of vertices, and a list of faces. There is no
actual requirement for those faces to be connected (Although in most applications, such a mesh would not be very
desirable!!).
1.6.5.2 Combining Mesh Geometry
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This component does not weld mesh vertices together, so it is often useful to use this in combination with a Weld
component.
Mesh Boolean
Meshes in Grasshopper have a set of boolean operations similar to boolean operations for NURBS solids.
Boolean operations are order specific, meaning that switching the order of the input meshes A and B will result in
different outputs.
1. Mesh Difference
2. Mesh Intersection
3. Mesh Split
4. Mesh Union
Intersect
Intersections can be calculated between meshes and other objects: rays, planes, curves, and other meshes
1.6.5.3 Intersections and Occlusions
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1. Mesh | Ray
2. Mesh | Plane
3. Mesh | Curve
4. Mesh | Mesh
Occlusion
As we have discussed, one of the (many) uses of mesh geometry is for visualizations and creating shaded
rendering based on face normals. When rendering, it also necessary to know when an object is in shadow behind
another object. The Occlusion component in Grasshopper allows us to enter a set of sample points, along with
occluding mesh geometry that will 'cast shadows', and a view ray, or vector, to indicate the direction that 'light' is
coming from.
Such a process can be used to create shadows in rendering, or determine whether objects are hidden from a
certain camera view.
1. View Ray to test for occlusion
2. Occluding mesh geometry
3. 'Hit' sample points
4. 'Occluded' sample points
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Since this definition is somewhat longer than previous examples in this primer, we will first walk through the basic
steps we will take:
1. Create a series of circles as a base cylinder
2. Use a Graph Mapper component to define the profile of our vase
3. Construct the topology of the mesh faces to produce a single mesh surface
4. Cap the bottom of the mesh
5. Introduce a twist to the vertical orientation for a more dynamic form
6. Add corrugated ridges for a textured vase
7. Offset the mesh surface to give the vase thickness
8. Cap the top gap between the two surfaces to produce a closed solid
01. Start a new definition, type Ctrl-N (in Grasshopper)
02. Params/Geometry/Point - Drag and drop a Point container onto the canvas
03.
Reference a point in Rhino by right-clicking the Point component and selecting "Set one
point". This will serve as the origin point of our vase. 
You can create a point manually in Grasshopper by double-clicking the canvas to
bring up the search window, then typing the coordinates of the point separated by
commas, such as: '0,0,0' (without quotes)
04.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Name: Length
Lower Limit: 1
Upper Limit: 10
05. Curve/Primitive/Line SDL - Drag and drop a Line SDL component onto the canvas
06.
Connect the Point component to the Start (S) input of the Line SDL component, and
connect the Number Slider to the Length (L) input. 
The default Direction (D) value of **Line SDL** is the Unit Z vector, which is what we
will use for this example
Params/Input/Number Slider - Drag and drop a Number Slider onto the canvas and set
the following values:
Name: V Count
1.6.6 Working with Mesh Geometry
In this section, we will work through an exercise file for producing a complete mesh solid. By the end of this
exercise, we will have a dynamic definition to produce custom vases that can be 3D printed.
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http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
07. Rounding: Integer
Lower Limit: 1
Upper Limit: 100
08. Curve/Division/Divide Curve - Drag and drop a Divide Curve component onto the canvas
09. Connect the Line (L) output of the Line SDL component to the Curve (C) input of the DivideCurve component
10. Connect the V Count number slider to the Count (N) input of the Divide Curve component
11. Curve/Primitive/Circle CNR - Drag and drop a Circle CNR component onto the canvas
12. Connect the Points (P) output of the Divide Curve component to the Center (C) input of theCircle CNR component
We have a series of circles stacked vertically. We will use these to make the profile of our vase.
Next, we will use a Graph Mapper to control the radii of the circles.
13. Sets/Sequence/Range - Drag and drop a Range component onto the canvas
14. Connect the V Count number slider to the Steps (N) input of the Range component
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15. Params/Input/Graph Mapper - Drag and drop a Graph Mapper component onto the
canvas
16. Right-click the Graph Mapper, click 'Graph Types' from the menu and select 'Bezier'
17.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Name: Width
Lower Limit: 0
Upper Limit: 10
18. Maths/Operators/Multiplication - Drag and drop a Multiplication component onto thecanvas
19. Connect the Graph Mapper and the Width number slider to the A and B inputs of theMultiplication component
20. Connect the Result (R) output of the Multiplication component to the Radius (R) input ofthe Circle CNR component
Use the handles on the Graph Mapper to adjust the profile of the circles.
NOTE: It is important to make sure the start point of the bezier curve on the Graph Mapper is not at zero. By
lifting the start point to a number greater than zero, we produce a flat base for our vase.
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We now have a profile for our vase. Next, we will construct a mesh surface. This will require creating mesh vertices
and defining mesh faces according to the index of those vertices.
21.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Name: U Count
Rounding: Even
Lower Limit: 2
Upper Limit: 100
22. Curve/Division/Divide Curve - Drag and drop a Divide Curve component onto the canvas
23.
Connect the Circle (C) output of the Circle CNR component to the the Curve (C) input of the
Divide Curve component, and connect the U Count number slider to the Count (N) input 
The Points(P) output of this component are the vertices we will use for our mesh
24. Sets/Sequence/Series - Drag and drop two Series components onto the canvas
25. Connect the U Count number slider to the Step (N) input of the first Series component, andconnect the V Count number slider to the Count (C) input of the same Series component
26. Connect the Series (S) output of the first Series component to the Start (S) input of thesecond Series component, and connect the U Count number slider to the Count (C) input
27. Sets/List/Shift List - Drag and drop a ShiftList component onto the canvas
28. Connect the output of the second Series component to the List (L) input of the Shift Listcomponent
29. Maths/Operators/Addition - Drag and drop two Addition components onto the canvas
30. Connect the output of the second Series component and the U Count number slider to theA and B inputs of the first Addition component
31. Connect the output of the Shift List component and the U Count number slider to the A andB inputs of the second Addition component
32. Mesh/Primitive/Mesh Quad - Drag and drop a Mesh Quad component onto the canvas
33.
Connect the following to the inputs of the Mesh Quad component:
A - Second **Series** component
B - **Shift List** 
C - First **Addition** component 
D - Second **Addition** component
We have just create the initial topology for our mesh. These faces will be combined
with the vertices. The order of these connections is crucial, so go ahead and double
check all the connections at this point!
34. Sets/Tree/Flatten - Drag and drop a Flatten Tree component onto tha canvas
35. Connect the Points (P) output of the Divide Curve component to the Tree (T) input of theFlatten Tree Component
36. Mesh/Primitive/Construct Mesh - Drag and drop a Construct Mesh component onto thecanvas
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37. Connect the Tree (T) output of the Flatten Tree component to the Vertices (V) input of theConstruct Mesh component
38. Connect the Face (F) output of the Mesh Quad component to the Faces (F) input of theConstruct Mesh component. Right-click the F (Faces) input and select 'Flatten'
We now have a mesh surface for our vase.
Next we will close the bottom of the vase. To do this, we will add the original origin point to our list of vertices, and
then construct triangle mesh faces from the bottom edge to that point.
39. Sets/Sequence/Series - Drag and drop a Series component onto the canvas
40. Connect the U Count number slider to the Count (C) input of the Series component
41. Sets/List/List Length - Drag and drop a List Length component onto the canvas
42.
Connect the Tree (T) output of the Flatten Tree component to the List (L) input of the List
Length component 
This will be the index of the origin point after we add it to the existing list of vertices.
43. Sets/List/Shift List - Drag and drop a Shift List component onto the canvas
Mesh/Primitive/Mesh Triangle - Drag and drop a Mesh Triangle component onto the
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44. canvas
45.
Connect the following to the inputs of the Mesh Triangle component:
A - Newest **Series** component
B - **List Length**
C - **Shift List**
46. Sets/Tree/Merge - Drag and drop two Merge components onto the canvas
47. Connect the Tree (T) output of the Flatten Tree component to the D1 input, and connect theinitial Point component to the D2 input of the first Merge component
48. Connect the Faces (F) of the Mesh Quad component to the D1 input, and connect theMesh Triangle output to the D2 input of the second Merge component
49.
Connect the first Merge component to the Vertices (V) input of the Construct Mesh
component, and connect the second Merge component to the Faces (F) input of the
Construct Mesh component.
We have capped the bottom of the vase with triangle mesh faces.
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We will now add some detailing to the vase. We will start by adding a curve to the vertical direction by adjusting the
seam of the original circles
50. Curve/Util/Seam - Drag and drop a Seam component onto the canvas
51. Connect the Circle (C) output of the Circle CNR component to the Curve (C) input of theSeam component
52. Right click the Curve (C) input of the Seam component and select 'Reparameterize'
53. Params/Input/Number Slider - Drag and drop a Number Slider component onto thecanvas. We will use the default settings for this slider
54. Maths/Operator/Multiplication - Drag and drop a Multiplication component onto thecanvas.
55. Connect the output from the Graph Mapper to the A input, and the newest Number Slider tothe B input of the Multiplication component
56. Connect the Result (R) of the Multiplication component to the Parameter (t) input of theSeam component
The curvature is achieved by changing the seam position of the initial circles, and uses the same Graph
Mapper as the vase profile.
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Next we will add some vertical ridges to the vase.
57. Sets/List/Dispatch - Drag and drop a Dispatch component onto the canvas
58.
Connect the Point (P) output of the second Divide Curve component to the List (L) input of
the Dispatch component 
We are using the default Pattern (P) input of the **Dispatch** component to separate
the points into two lists with alternating points
59. Vector/Vector/Vector 2Pt - Drag and drop a Vector 2Pt component onto the canvas
60. Connect the B output of the Dispatch component to the A input of the Vector 2Ptcomponent
61. Connect the Points (P) of the first Divide Curve component to the B input of the Vector 2Ptcomponent
62.
Right-click the B input of the Vector 2Pt component and select 'Graft', and right-click the
Unitize (U) input, go to 'Set Boolean' and select 'True' 
This creates a unit vector for each point that points towards the center of the circle
63. Params/Input/Number Slider - Drag and drop a Number Slider component onto thecanvas. We will use the default settings
64. Maths/Operator/Multiplication - Drag and drop a Multiplication component onto thecanvas
65. Connect the Vector (V) output of the Vector 2Pt component to the A input, and connect theNumber Slider to the B input of the Multiplication component
66. Transform/Euclidean/Move - Drag and drop a Move component onto the canvas
67. Connect the B output of the Dispatch component to the Geometry (G) input of the Movecomponent
68. Connect the Result (R) output of the Multiplication component to the Motion (T) input of theMove component
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69. Sets/List/Weave - Drag and drop a Weave component onto the canvas
70. Connect the A output of the Dispatch component to the 0 input of the Weave component
71. Connect the Geometry (G) output of the Move component to the 1 input of the Weavecomponent
72. Connect the Weave (W) output of the Weave component to the Tree (T) input of the FlattenTree component
Remember to go back and adjust your sliders and graph mapper to see how the model changes, and to
make sure everything still works. This is known as 'flexing' the model, and should be done frequently to check
for mistakes in the definition.
We now have a single surface for our vase. If we wanted to print this vase using a 3D printer, we need it to be a
closed solid. We will create a solid by offsetting the current mesh, then combining the original mesh and the offset
mesh.
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73. Mesh/Analysis/Deconstruct Mesh - Drag and drop a Deconstruct Mesh component ontothe canvas
74. Connect the Mesh (M) output of the Construct Mesh component to the Mesh (M) input ofthe Deconstruct Mesh component
75. Params/Input/Number Slider - Drag and drop a Number Slider component onto thecanvas. We will use the default settings
76. Maths/Operator/Multiplication - Drag and drop a Multiplication component onto thecanvas
77. Connect the Normals (N) output of the Deconstruct Mesh component to the A input, andconnect the Number Slider to the B input of the Multiplication component
78. Transform/Euclidean/Move - Drag and drop a Move component onto the canvas
79. Connect the Vertices (V) output of the Deconstruct Mesh component to Geometry (G) inputof the Move component
80. Connect the Result (R) output of the Multiplication component to the Motion (T) input of theMove component
81. Mesh/Primitive/Construct Mesh Drag and drop a Construct Mesh component onto thecanvas
82. Connect the Geometry(G) output of the Move component to the Vertices (V) input of theConstruct Mesh component
83. Connect the Faces (F) output of the Deconstruct Mesh component to the Face (F) input ofthe Construct Mesh component
By offseting the mesh according to the vertex normals, we now have an 'inside' and an 'outside' mesh, but we
still have a gap at the top between the two mesh geometries
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The final step will be to create a closed mesh by creating a new mesh geometry to close the gap and then joining
the meshes together.
84. Mesh/Analysis/Mesh Edges - Drag and drop a Mesh Edges component onto the canvas
85. Connect the Mesh (M) output of the first Construct Mesh component to the Mesh (M) inputof the Mesh Edges component
86. Curve/Util/Join Curves - Drag and drop a Join Curves component onto the canvas
87. Connect the Naked Edges (E1) output of the Mesh Edges component to the Curves (C)input of the Join Curves component
88. Curve/Analysis/Control Points - Drag and drop a Control Points component onto thecanvas
89.
Connect the Curves (C) output of the Join Curves component to the Curve (C) input of the
Control Points component 
By joining the curves and then extrating the control points, we ensure that the order
of the points is consistent along the rim of the vase, which is important for making
the resulting mesh orientable and manifold
90. Sets/List/Shift List - Drag and drop a Shift List component onto the canvas
91. Connect the Points (P) output of the Control Points component to the List (L) input of theShift List component
92. Repeat steps 84 through 91 for the second Construct Mesh component
93. Sets/Tree/Entwine - Drag and drop an Entwine component onto the canvas
94.
Zoom in to the Entwine component to show the option to add an extra input. We will need
four inputs. Connect them in the following way:
{0;0} - Points (P) from first **Control Points** component
{0;1} - output from first **Shift List** 
{0;2} - output from second **Shift List**
{0;3} - Points (P) from second **Control Points** component
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95. Sets/Tree/Flip Matrix - Drag and drop a Flip Matrix component onto the canvas
96. Connect the Result (R) from the Entwine component to the Data (D) input of the FlipMatrix component
97. Mesh/Primitive/Construct Mesh - Drag and drop a Construct Mesh component onto thecanvas
98. Connect the Data (D) outut of the Flip Matrix component to the Vertices (V) input of theConstruct Mesh component
99. Mesh/Util/Mesh Join - Drag and drop a Mesh Join component onto the canvas
100.
Connect all three Construct Mesh components to the Mesh Join component by holding
down the Shift key while connecting the wires (or use a Merge component). Right-click
the Mesh (M) input of the Mesh Join component and select 'Flatten'
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2. Extensions
Foundations are meant to be built upon. This volume features an array of key
plugins for Grasshopper that will extend the application's functionality and your
ability to take your designs further.
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192Extensions
2.1. Element*
Element* is a mesh geometry plugin for Grasshopper, enabling mesh creation,
analysis, transformation, subdivision, and smoothing. Element* provides access
to mesh topology data using the Plankton half-edge data structure for polygon
meshes.
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193Element*
Download the Element* plug-in to get started
2.1.1. Element*
Integrating the use of Meshes in your workflow offers a wide variety of opportunities to create shapes that
range from faceted to smooth. Element* allows you to go further, by giving you more intuitive access to
analyzing mesh topology and smoothing routines. This chapter is a User's Guide for Element* Version 1.1 and
will get you up to speed.
Element* components are categorized based on their operations. Much like the periodic table, which provides a
framework for analyzing chemical behavior, Element provides a framework for analyzing and exloring
geometry based on mesh data and operations. We imagine new components will be created by analysing the
relationships between components in each category.
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194Introduction
http://www.food4rhino.com/project/element
Below are some inspirational images of the types of products and applications that could be generated using
Element*.
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The Grasshopper Primer V3.3
196Introduction
The Grasshopper Primer V3.3
197Introduction
In the Grasshopper primer, we looked at how Grasshopper defines a mesh using a Face-Vertex data structure. This
is a relatively simple data structure and is widely used in applications that use meshes, but can be computationally
inefficient for more advanced algorithms. The Element* add-on restructures the mesh using Half-Edge data, an
edge-centered data structure, which allows for efficient queries of adjacent vertices, faces, and edges, which can
vastly improve on algorithm speed and performance. This structure is capable of maintaining incidence information
of vertices, edges and faces. This method facilitates the creation of new patterns and geometries all based on the
topological relationship of the base geometry.
The half-edge data structure is a representation for a mesh in which each edge is split up into two half-edges with
opposite directions. This allows explicit and implicit access to data from one mesh element to adjacent elements.
The half-edge highlighted in blue explicitly stores indices to its termination point, adjacent half-edges, and the face
it belongs to. The other information (gray) can be accessed implicitly.
2.1.2. Half Edge Data
2.1.2.1 Half-Edge Connectivity
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The vertex highlighted in blue explicitly stores an index to one of its outgoing half-edges. The other information
(gray) can be accessed implicitly.
2.1.2.2 Vertex Connectivity
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1. Mesh Closest Point
2. Mesh Evaluate
3. Mesh Sample Plus
Element* Mesh Closest Point
Unlike Grasshopper's Mesh Closest Point component, this component also calculates the normal and color at the
outputed point, eliminating the need for a Mesh Eval component and simplfying the canvas workspace.
Element* Mesh Evaluate
The built in Grasshopper Mesh Eval component requires a mesh parameter as an input, which can be extracted
from a Mesh Closest Point component, but which can be difficult to construct manually. Element's closest point
component allows direct input of the index of a mesh face and the barycentric coordinates.
Note - barycentric coordinates are defined such that they always add to 1. If the input values of U,V, and W do not
add to 1, this component will maintain the ratio of the three values while normalizing them. For example, if you had
the input values of 2, 2, and 4 the mesh parameter would be calculated as {0.25;0.25;0.5}
Element* Mesh Sample Plus
This component is used to quickly extract color information from a mesh. It returns the Alpha, Red, Green, Blue,
Hue, Saturation, and Luminosity values of the inputed points. If the given points are not on the mesh, this
component will calculate the closest point. This component uses Parallel Computing for speed.
2.1.3. Element* Components
2.1.3.1 Analyse
2.1.3.2 Data
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1. Data Visualizer
2. Edge Neighbors
3. Face Neighbors
4. Vertex Neighbors
Element* Data Visualizer
This component is used to help visualize the half-edge data of the faces of an input mesh.
Element* Edge Neighbors
This component provides access to the adjacency data structured according to the edges of the input mesh. The
output data is provided as a tree with one branch for each edge in the mesh. It returns the mesh edges, the edge
end points,center ponts of the faces adjacent to each edge (dual), the neighboring edges as line objects (arranged
in clockwise order) , and neighbouring face centers (center points of faces adjacent to the edge start and end
points)
Edge Neighbors - Edges, End vertices, Adjacent face centers, Neighboring edges, and Neighbouring face
centers
Element* Face Neighbors
This component is similar to the others in this section, but the data is organized in a tree according to the faces of
the mesh, with one branch per face. The outputs are the face centers, vertices of each face (arranged in counter
clockwise order), neighbouring edges (arranged in counter clockwise order), and the centers of neighboring faces
(arranged in counter clockwise order).
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Face Neighbors - Face centers, face vertices, neighbouring edges, neighbouring face centers
Element* Vertex Neighbors
This component outputs the mesh vertices, neighboring vertices (arranged in clockwise order), neighbouring
edges (arranged in clockwise order), and neighbouring face centers (arranged in clockwise order) structured in a
tree according to the vertices of the mesh.
Vertex Neighbors - Vertices, neighbouring vertices, neighbouring edges, neighbouring face centers
Element* provides four additional mesh primitives: the Dodecahedron, Tetrahedron, Octahedron, and Icosahedron.
These components take a single number as input for the radius, and produce meshes centered at the origin, and
composed of one face per side. With the addition of the Cube, which is already availible through Grasshopper's
built-in primitives, these make up the five Platonic solids.
1. Dodecahedron
2.1.3.3 Primitives
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2. Tetrahedron
3. Octahedron
4. Icosahedron
Element* Smooth provides an optimized smoothing algorithm that is more efficient than Grasshopper's Smooth
Mesh for large datasets. It makes use of the Lapacian Smoothing algorithm for Half-Edge structured meshes. It
does not change the topology or vertex count of welded meshes, but will combine identical vertices if there are any
duplicates caused by an unwelded mesh. We can specify the smoothing strength, boundary condition, boundary
tolerance as well as the number of iterations.
Element* Catmull Clark Subdivision
This is a recursive subdivision defined by the Catmull Clark algorithm. We can specify the number of iterations as
well as how to handle naked edge conditions.
Element* Constant Quad
This subdivison component will create an all quad mesh by adding a face for each edge of the mesh.
2.1.3.4 Smooth
2.1.3.5 Subdivide
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1. Constant Quad subdivision
2. Catmull Clark subdivision
1. Mesh Windown
2. Mesh Frame
3. Mesh Thicken
4. Mesh Offset
5. Mesh Poke Face
These components provide a number of different transformations described below. Each component has the
additional capability of accepting per-vertex distance data to allow for variations of the transformation magnitudes
across the mesh.
Element* Mesh Window
Reconstructs a new mesh on the inside of a face based on an offset value. This component accepts either a mesh
or a list of closed polylines as input.
Element* Mesh Frame
Outputs a frame around mesh faces. Each resultant face will have a new hole in the center. This component
accepts either a mesh or a list of closed polylines as input.
Element* Mesh Thicken
This component will thicken an input mesh along the vertex normals, and according to provided distance values.
Element* Mesh Offset
This component creats an offset of the input mesh based on the vertex normals.
Element* Mesh Poke Face
First the mesh face goes through the frame operation then the face inner is split the selected faces and allows the
user to specify the push or pull amount from the center of the original polygon. For example, a four-sided polygon
(quad) is split into 4 three-sided polygons with one shared vertex in the middle. The height input allows you to
transform that vertex.
2.1.3.6 Transform
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1. Mesh Window
2. Mesh Frame
3. Icosohedron transformed with mesh frame, then thickend and subdivided
1. Mesh Combine & Clean
2. Mesh Edges
3. Mesh Status
The Element* Mesh Combine and Clean
This component combines multiple meshes, has options for either welding a mesh based on input angle or
combining identical vertices. We can also flip the orientation of the mesh. This component also detects potential
topology issues and returns Remarks and Warnings with detailed explanantions. In the event that combining
identical vertices creates bad topology the component will return the input list of meshes instead of a combined
merged mesh. The user can also choose to combine the mesh without merging any of its vertices.
The Element* Mesh Edges
This component returns the mesh naked edges, mesh edges, face polylines and if the mesh is unwelded it will
return the unwelded mesh edges.
The Element* Mesh Status
This component returns mesh information based on the topology. There are two modes which we can view the
information, the first is Mesh Info which returns Geometry data such as Mesh Validity, Vertex Count, Face Count, and
2.1.3.7 Utility
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Normal Count. The other returns the Mesh Status, which is the condition of the mesh, whether it has non manifold
edges, degenerate face count, naked edge count, and disjoined mesh count. This component does not operate on
a mesh it simply returns the information to the user. There is also an option to combine identical vertices, therefore
the user can see the effects this would have on the mesh.
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
01. Start a new definition, type Ctrl-N (in Grasshopper)
02. Element*/Primitive/Icosohedron - Drag and drop the Icosohedron component onto thecanvas
03. Params/Input/Number Slider - Drag and drop the Number Slider component onto thecanvas
04. Connect the Number Slider to the Radius (R) input of the Icosohedron component
05.
Double-click the Number Slider and set appropriate values. For this example, we used:
Name: Radius
Rounding: Integer
Lower Limit: 5
Upper Limit: 50
Value: 25
06. Element*/Data/Face Neighbors - Drag and drop the Face Neighbors component onto thecanvas
07. Connect the Mesh (M) output of the Icosohedron component to the Mesh (M) input of theFace Neighbors component.
2.1.4. Exercise
In this section, we will work through a simple exercise using the Element* primitives as a base. We will
incorporate the half-edge data structure as well using both features of the transform components (uniform and
per vertex)
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http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
Looking at the data of the Neighboring Face Edges (NE) output, we see that we have a tree with 20 branches,
where each branch contains three lines. The 20 branches each represent a face of the icosohedron which
has 20 sides, while the three lines are the edges of each triangular face.
08.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Rounding: Float
Lower Limit:0
Upper Limit: 0.5
09. Params/Input/Panel - Drag and drop a Panel component onto the canvas
10. Double-click the Panel component and enter "1" into the text-field
11. Math/Operators/Subtraction - Drag and drop a Subtraction component onto the canvas
12. Connect the Panel with a value of "1" into the A input and connect the number slider to theB input of the Subtraction component
13. Sets/Tree/Merge - Drag and drop a Merge component onto the canvas
14. Connect the Number Slider to the D1 input of Merge, and connect the output R of theSubtraction component to the D2 input of Merge
15. Curve/Analysis/Evaluate Curve - Drag and drop an Evaluate Curve component onto thecanvas
16.Connect the Face Edges (NE) output of the Face Neighbors component to the Curve (C)input of the the Evaluate Curve component
17. Right click the Curve (C) input of the Evaluate Curve component and select Graft. This willcreate a new branch for each edge.
18.
Connect the Result (R) output of the Merge component to the Parameter (t) input of the
Evaluate Curve component. Because we grafted the Curve input, each edge is evaluated
at both parameters from Merge
19. Sets/Tree/Trim Tree - Drag and drop a Trim Tree component onto the canvas
Connect the Points (P) output of Evaluate Curve to the Tree (T) input of the Trim Tree
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20.
component. 
The default value of Depth (D) input for **Trim Tree** is 1. This reduce the depth of
our data tree one level by merging the outer most branch. The result is 20 branches,
each with six points.
21. Curve/Spline/Polyline - Drag and drop a Polyline component onto the canvas
22. Connect the Tree (T) output of the Trim Tree component to the Vertices (V) input of thePolyline component
23.
Right click the Closed (C) input of the Polyline component, click "Set Boolean" and set the
value to True 
This has created a closed polyline of six sides for each original face of the mesh.
24. Element*/Transform/Mesh Frame - Drag and drop a Mesh Frame component onto thecanvas.
25.
Connect the Polyline (Pl) output of the Polyline component to the Geometry (G) input of the
Mesh Frame component 
Note that the **Mesh Frame** component can take either meshes or a list of closed
polyline curves as input
26. Params/Input/Number Slider - Drage and drop a Number Silder component onto thecanvas. We will keep the default range of 0 to 1 for this slider
27. Connect the Number Slider to the Factor (F) input of the Mesh Frame component
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28. Element*/Utility/Mesh Combine and Clean - Drag and drop a Mesh Combine and Clean
component on the canvas
29. Connect the Mesh (M) output of Mesh Frames to the Mesh (M) input of the Mesh Combineand Clean component
30.
Right click the Mesh (M) input of Mesh Combine and Clean and select Flatten 
By flattening the tree of meshes, **Combine and Clean** will merge all 20 face
meshes into a single mesh
31. Element*/Transform/Mesh Thicken - Drag and drop a Mesh Thicken component onto thecanvas
32. Connect the Mesh (M) output of Combine and Clean to the Mesh (M) input of Mesh Thicken
33. Element*/Subdivide/Catmull Clark Subdivision - Drag and drop a Catmull ClarkSubdivision component onto the canvas
34. Connect the Mesh (M) output of Mesh Thicken to the Mesh (M) input of the Catmull ClarkSubdivision component
We have truncated the triangular faces of the initial mesh, effectively also creating rings around each original
vertex. We have also created a frame for each face, then thickened the mesh and refined it with subdivision.
Next we will take advantage of the Per Vertex capabilities of the transform components by using an attractor
point.
35. Params/Geometry/Point - Drag and drop a Point parameter onto the canvas
Right click the Point parameter and select "Set on point" to select a point in the Rhino
viewport 
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36. Tip - you can also create a point directly in Grasshopper by double-clicking the
canvas to bring up the Search window, then typing a point coordinate such as
"10,10,0" (without the quotes)
37. Mesh/Analysis/Deconstruct Mesh - Drag and drop a Deconstruct Mesh component ontothe canvas
38.
Connect the Mesh (M) output of the Combine and Clean component to the Mesh (M) input
of the Deconstruct Mesh component. 
We will use this to extract the vertices of our combined mesh, and then apply an
attractor point to these vertices
39. Vector/Point/Distance - Drag and drop a Distance component onto the canvas
40. Connect the Vertices (V) output of the Deconstruct Mesh component to the A input of thethe Distance component
41. Connect the Point parameter to the B input of the Distance component
42. Connect the Distance (D) output of the Distance component to the PerVectex Data (VD)input of the Thicken component
43.
Params/Input/Number Slider - Drag and drop two Number Slider components onto the
canvas. We will use these to set the lower and upper limits for the Mesh Thicken
component
44. Double-click the Number Sliders and set the values. In this example, we left the first sliderat default values, and set the Upper Limit of the second slider to 5.0
45. Maths/Domain/Construct Domain - Drag and drop a Construct Domain component ontothe canvas
46. Connect the two number sliders to the A and B inputs of the Construct Domain component
47. Connect the Domain (I) output of the Construct Domain component to the Min and MaxValues (D) input of the Mesh Thicken component.
48.
Right click the Type (T) input of the Thicken component, select "Set Integer" and enter a
value of 1 
You can also enable the PerVertex Data by using a **Boolean Toggle** component
set to True.
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Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
2.1.5. Element* Architectural Case Study
In this section, we will work through a simple exercise file that is meant as an introduction to working with
Element tools. We will explore some patterning and facade treatments in the field of Architecture which will
incorporate Half Edge data structures along with basic Element components without the use of per vertex
features.
2.1.5.1 Example 1
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http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
00. Create a meshplane in Rhino with XFaces = 2 & YFaces = 2 and Start a new definition,type Ctrl-N (in Grasshopper)
01. Params/Geometry/Mesh - Drag and drop a Mesh container onto the canvas
01b.
Reference a mesh in Rhino by right-clicking the Mesh component and selecting "Set one
Mesh". 
We are going to use a simple mesh plane to walk through the definition, feel free to
swap out the mesh with your own mesh
02. Element*/Utility/Mesh Combine and Clean - Drag and drop a Element* Mesh Combineand Clean component on the canvas
03. Element*/Data/Vertex Neighbors - Drag and drop the Element* Vertex Neighborscomponent onto the canvas
04.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Lower Limit: 0.0000
Upper Limit: 1.0000
05. Curve/Analysis/Evaluate Curve - Drag and drop a Evaluate Curve container onto thecanvas
05b. Connect the Neighbouring Edges (NE) output of the Element* Vertex Neighborscomponent to the Curve (C) input of the Evaluate Curve component
05c. Connect the Number Slider to the Float (t) input of the Evaluate Curve component and setthe value to 0.5000
05d. Right click the Curve (C) input of the Evaluate Curve component and enableReparameterize
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06. Element*/Analyse/Mesh Closest Point - Drag and drop a Element* Mesh Closest Pointcontainer onto the canvas
06a. Connect the Mesh output (M) of the Element*/Utility/Mesh Combine and Cleancomponent to the Mesh (M) input of the Element* Mesh Closest Point component
06b. Connect the Points output (P) of the Curve/Analysis/Evaluate Curve component to thePoint (P) input of the Element* Mesh Closest Point component
07.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Rounding: Float
Lower Limit:0
Upper Limit: 10.000
08. Vector/Vector/Amplitude - Drag and drop a Amplitude component onto the canvas
09. Transform/Euclidean/Move - Drag and drop a Move component onto the canvas
10. Params/Geometry/Point - Drag and drop a Point container onto the canvas
10b. Connect the Face Centers output (FC) of the Element* Vertex Neighbors component tothe Point component
11. Sets/List/Weave - Drag and drop a Weave componentonto the canvas
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12. Curve/Primitive/Polyline - Drag and drop a Polyline component onto the canvas
12b. Connect the Weave output (W) of the Weave component to the Vertices (V) input of thePolyline component
12c.
Right click the Closed (C) input of the Polyline component, click "Set Boolean" and set the
value to True 
This has created a closed polyline.
13. Params/Input/Number Slider - Drage and drop a Number Silder component onto thecanvas. We will keep the default range of 0 to 1 for this slider
14. Element*/Transform/Mesh Frame - Drag and drop a Element* Mesh Frame componentonto the canvas.
14b.
Connect the Polyline (Pl) output of the Polyline component to the Geometry (G) input of
the Mesh Frame component 
Note that the **Mesh Frame** component can take either meshes or a list of
closed polyline curves as input
14c. Connect the Number Slider to the Factor (F) input of the Mesh Frame component
15. Element*/Utility/Mesh Combine and Clean - Drag and drop a Element* Mesh Combineand Clean component on the canvas
Right click the Combine Type (CT) input of the Element* Mesh Combine and Clean
component, click "Set Integer" and set the value to 1. 
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15b. The Combine Type input has two options (0, which combines and cleans the
meshes) and (1, which joins the meshes in the list without merging vertices). In
this example we want to join the meshes
16.
Right click the Mesh (M) input of the Element* Mesh Combine and Clean component, click
"Flatten". 
This will flatten the list so we can join the mesh list together.
17. Element*/Utility/Mesh Status - Drag and drop a Element* Mesh Status component onthe canvas
17b
Connect the Info (I) and Status (S) outputs of Element* Mesh Status to a
Params/Input/Panel component 
The mesh **Info** output contains mesh validity information, closed or open type
and mesh component counts (vertices, faces, normals). The mesh **Status**
informs the user if the mesh is in "Good" condition as well as data regarding non
manifold edges, unused vertex count, degenerate face count, naked edge count
and disjoined mesh count.
18. Params/Input/Colour Swatch - Drag and drop a Colour Swatch component on thecanvas
19. Display/Preview/Custom Preview - Drag and drop a Custom Preview component on thecanvas
Example files that accompany this section: http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
2.1.5.2 Example 2
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217Architectural Case Study
http://grasshopperprimer.com/appendix/A-2/1_gh-files.html
00. Create a meshplane in Rhino with XFaces = 2 & YFaces = 2 and Start a new definition,type Ctrl-N (in Grasshopper)
01. Params/Geometry/Mesh - Drag and drop a Mesh container onto the canvas
01b.
Reference a mesh in Rhino by right-clicking the Mesh component and selecting "Set one
Mesh". 
We are going to use a simple mesh plane to walk through the definition, feel free to
swap out the mesh with your own mesh
02. Element*/Utility/Mesh Combine and Clean - Drag and drop a Element* Mesh Combineand Clean component on the canvas
03. Element*/Data/Vertex Neighbors - Drag and drop the Element* Vertex Neighborscomponent onto the canvas
04. Vector/Vector/Vector2Pt - Drag and drop a Vector2Pt component onto the canvas
05.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Rounding: Float
Lower Limit:0
Upper Limit: 2.000
06. Maths/Operator/Multiplication - Drag and drop a Multiplication component onto thecanvas
07. Maths/Operators/Addition - Drag and drop two Addition components onto the canvas
08. Curve/Primitive/Polyline - Drag and drop a Polyline component onto the canvas
09. Curve/Primitive/Polyline - Drag and drop a Polyline component onto the canvas
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10.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Rounding: Float
Lower Limit:0
Upper Limit: 1.000
11,12. Element*/Transform/Mesh Frame - Drag and drop a Element* Mesh Framecomponent onto the canvas.
11b,12b.
Connect the Polyline (Pl) output of the Polyline component to the Geometry (G) input
of the Mesh Frame component 
Note that the **Mesh Frame** component can take either meshes or a list of
closed polyline curves as input
11c,12c. Connect the Number Slider (10) to the Factor (F) input of the Mesh Frame component
13,14.
Element*/Subdivide/Catmull Clark Subdivision - Drag and drop a Catmull Clark
Subdivision component onto the canvas 
We will set the Iterations input (I) value to 1 as well as the **Edge Condition**
input (E) to a value of 1. The edge condition input options are 0 = Fixed, 1 ==
Smooth, 2 == Corners Fixed.
15. Sets/Tree/Merge - Drag and drop two Merge components onto the canvas
15b. Right click the Result (R) output of the Merge component and click "Flatten".
16. Element*/Utility/Mesh Combine and Clean - Drag and drop a Element* MeshCombine and Clean component on the canvas
Components have detailed remarks and warnings to inform the user of the current or potential issues that
might come about from interaction with other components. In some instances you might use the Element
Combine and Clean component to join and combine identical vertices on a mesh which could lead to non
manifold edges if that mesh is thickend later on. The Element Combine and Clean component will inform you
of this issue that will return the list back to you. You have the option of setting the Combine Type to a value of 1
which will combine the meshes in the list but not combine identical vertices.
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17. Element*/Utility/Mesh Edges - Drag and drop a Element* Mesh Edges component on thecanvas
17b Connect the Mesh (M) output of the Element* Mesh Combine and Clean component (16)to the Mesh input (M) of the Element* Mesh Edges component
18.
Params/Input/Number Slider - Drag and drop a Number Slider component onto the
canvas and set the following values:
Rounding: Float
Lower Limit:0
Upper Limit: 1.000
19. Element*/Transform/Mesh Frame - Drag and drop a Element* Mesh Frame componentonto the canvas.
19b Connect the Face Polylines (FP) output of the Element* Mesh Edges component to theMesh input (M) of the Element* Mesh Frame component
19c Connect the Number Slider to the Float (f) input of the Element* Mesh Frame component
20. Element*/Utility/Mesh Combine and Clean - Drag and drop a Element* Mesh Combineand Clean component on the canvas
21. Right click the Mesh (M) input of the Element* Mesh Combine and Clean component andclick "Flatten".
Right click the Combine Type (CT) input of the Element* Mesh Combine and Clean
The Grasshopper Primer V3.3
220Architectural Case Study
22.
component, click "Set Integer" and set the value to 1. 
The Combine Type input has two options (0, which combines and cleans the
meshes) and (1, which joins the meshes in the list without merging vertices). In this
example we want to join the meshes
23. Params/Input/Colour Swatch - Drag and drop a Colour Swatch component on thecanvas
24. Display/Preview/Custom Preview - Drag and drop a Custom Preview component on thecanvas
25. Element*/Utility/Mesh Status - Drag and drop a Element* Mesh Status component on thecanvas
25b
Connect the Info (I) and Status (S) outputs of Element* Mesh Status to a
Params/Input/Panel component 
The mesh **Info** output contains mesh validity information, closed or open type
and mesh component counts (vertices, faces, normals). The mesh **Status**
informs the user if the mesh is in "Good" condition as well as data regarding non
manifold edges, unused vertex count, degenerate face count, naked edge count
and disjoined mesh count.
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221Architectural Case Study
Appendix
The following section contains useful references including an index of all the
components used in thisprimer, as well as additional resources to learn more
about Grasshopper.
The Grasshopper Primer V3.3
222Appendix
The Grasshopper Primer V3.3
223Appendix
P.G.Crv Curve Parameter
Represents a collection of Curve geometry. Curve geometry is
the common denominator of all curve types in Grasshopper.
P.G.Circle Circle Parameter
Represents a collection of Circle primitives.
P.G.Geo Geometry Parameter
Represents a collection of 3D Geometry.
P.G.Pipeline Geometry Pipeline
Defines a geometry pipeline from Rhino to Grasshopper.
P.G.Pt Point Parameter
Point parameters are capable of storing persistent data. You can
set the persistent records through the parameter menu.
P.G.Srf Surface Parameter
Represents a collection of Surface geometry. Surface geometry
is the common denominator of all surface types in Grasshopper.
P.P.Bool Boolean Parameter
Represents a collection of Boolean (True/False) values.
P.P.D Domain Parameter
Represents a collection of one-dimensional Domains. Domains
are typically used to represent curve fragments and continuous
numeric ranges. A domain consists of two numbers that indicate
the limits of the domain, everything in between these numbers is
part of the domain.
P.P.D2 Domain2 Parameter
Contains a collection of two-dimensional domains. 2D Domains
are typically used to represent surface fragments. A two-
dimensional domain consists of two one-dimensional domains.
P.P.ID Guid Parameter
Represents a collection of Globally Unique Identifiers. Guid
parameters are capable of storing persistent data. You can set
the persistent records through the parameter menu.
P.P.Int Integer Parameter
Represents a collection of Integer numeric values. Integer
parameters are capable of storing persistent data. You can set
2.1. Index
This index provides additional information on all the components used in this primer, as well as other
components you might find useful. This is just an introduction to over 500 components in the Grasshopper
plugin.
Parameters
Geometry
Primitive
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the persistent records through the parameter menu.
P.P.Num Number Parameter
Represents a collection of floating point values. Number
parameters are capable of storing persistent data. You can set
the persistent records through the parameter menu.
P.P.Path File Path
Contains a collection of file paths.
P.I.Toggle Boolean Toggle
Boolean (true/false) toggle.
P.I.Button Button
Button object with two values. When pressed, the button object
returns a true value and then resets to false.
P.I.Swatch Color Swatch
A swatch is a special interface object that allows for quick setting
of individual color values. You can change the color of a swatch
through the context menu.
P.I.Grad Gradient Control
Gradient controls allow you to define a color gradient within a
numeric domain. By default the unit domain (0.0 ~ 1.0) is used,
but this can be adjusted via the L0 and L1 input parameters. You
can add color grips to the gradient object by dragging from the
color wheel at the upper left and set color grips by right clicking
them.
P.I.Graph Graph Mapper
Graph mapper objects allow you to remap a set of numbers. By
default the {x} and {y} domains of a graph function are unit
domains (0.0 ~ 1.0), but these can be adjusted via the Graph
Editor. Graph mappers can contain a single mapping function,
which can be picked through the context menu. Graphs typically
have grips (little circles), which can be used to modify the
variables that define the graph equation. By default, a graph
mapper objects contains no graph and performs a 1:1 mapping
of values.
P.I.Slider Number Slider
A slider is a special interface object that allows for quick setting of
individual numeric values. You can change the values and
properties through the menu, or by double-clicking a slider
object. Sliders can be made longer or shorter by dragging the
rightmost edge left or right. Note that sliders only have an output
(ie. no input).
P.I.Panel Panel
A panel for custom notes and text values. It is typically an inactive
object that allows you to add remarks or explanations to a
Document. Panels can also receive their information from
elsewhere. If you plug an output parameter into a Panel, you can
see the contents of that parameter in real-time. All data in
Grasshopper can be viewed in this way. Panels can also stream
their content to a text file.
P.I.List Value List
Input
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P.U.Cin Cluster Input
Represents a cluster input parameter.
P.U.COut Cluster Output
Represents a cluster input parameter.
P.U.Dam Data Dam
Delay data on its way through the document.
P.U.Jump Jump
Jump between different locations.
P.U.Viewer Param Viewer
A viewer for data structures.
P.U.Scribble Scribble
A quick note.
M.D.Bnd Bounds
Create a numeric domain which encompasses a list of
numbers.
M.D.Consec Consecutive Domains
Create consecutive domains from a list of numbers.
M.D.Dom Construct Domain
Create a numeric domain from two numeric extremes.
M.D.Dom2Num Construct Domain²
Create a two-dimensional domain from four numbers.
M.D.DeDomain Deconstruct Domain
Deconstruct a numeric domain into its component parts.
M.D.DeDom2Num Deconstruct Domain²
Deconstruct a two-dimensional domain into four numbers.
M.D.Divide Divide Domain²
Divides a two-dimensional domain into equal segments.
Utilities
Maths
Domain
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M.D.Divide Divide Domain²
Divides a two-dimensional domain into equal segments.
M.D.Inc Includes
Test a numeric value to see if it is included in the domain.
M.D.ReMap Remap Numbers
Remap numbers into a new numeric domain.
M.O.Add Addition
Mathematical addition.
M.O.Div Division
Mathematical division.
M.O.Equals Equality
Test for (in)equality of two numbers.
M.O.And Gate And
Perform boolean conjunction (AND gate). Both inputs need to be
True for the result to be True.
M.O.Not Gate Not
Perform boolean negation (NOT gate).
M.O.Or Gate Or
Perform boolean disjunction (OR gate). Only a single input has
to be True for the result to be True.
M.O.Larger Larger Than
Larger than (or equal to).
M.O.Multiply Multiplication
Mathematical multiplication.
M.O.Smaller Smaller Than
Larger than (or equal to).
M.O.Similar Similarity
Test for similarity of two numbers.
M.O.Sub Subtraction
Mathematical subtraction.
Operators
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M.S.Eval Evaluate
Evaluate an expression with a flexible number of variables.
M.S.Expression Expression
Evaluate an expression.
M.T.Cos Cosine
Compute the cosine of a value.
M.T.Deg Degrees
Convert an angle specified in radians to degrees.
M.T.Rad Radians
Convert an angle specified in degrees to radians.
M.T.Sin Sine
Compute the sine of a value.
M.U.Avr Average
Solve the arithmetic average for a set of items.
M.U.Phi Golden Ratio
Returns a multiple of the golden ratio (Phi).
M.U.Pi Pi
Returns a multiple of Pi.
S.L.Combine Combine Data
Combine non-null items out of several inputs.
S.L.CrossRef Cross Reference
Cross Reference data from multiple lists.
S.L.Dispatch Dispatch
Dispatch the items in a list into two target lists. List dispatching
is very similar to the [Cull Pattern] component, with the exception
that both lists are provided as outputs.
S.L.Ins Insert Items
Insert a collection of items into a list.
Script
Trig
Utilities
Sets
List
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S.L.Item List Item
Retrieve a specific item from a list.
S.L.Lng List Length
Measure the length of a list. Elements in a list are identified by
their index. The first element is stored at index zero, the second
element is stored at index one and so on and so forth. The
highest possible index in a list equals the length of the list
minus one.
S.L.Long Longest List
Grow a collection of lists to the longest length amongst them.
S.L.Split Split List
Split a list into separate parts.
S.L.Replace Replace Items
Replace certain items in a list.
S.L.Rev Reverse List
Reverse the order of a list. The new index ofeach element will
be N-i where N is the highest index in the list and i is the old
index of the element.
S.L.Shift Shift List
Offset all items in a list. Items in the list are offset (moved)
towards the end of the list if the shift offset is positive. If Wrap
equals True, then items that fall off the ends are re-appended.
S.L.Short Shortest List
Shrink a collection of lists to the shortest length amongst them.
S.L.Sift Sift Pattern
Sift elements in a list using a repeating index pattern.
S.L.Sort Sort List
Sort a list of numeric keys. In order for something to be sorted, it
must first be comparable. Most types of data are not
comparable, Numbers and Strings being basically the sole
exceptions. If you want to sort other types of data, such as
curves, you’ll need to create a list of keys first.
S.L.Weave Weave
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Weave a set of input data using a custom pattern. The pattern is
specified as a list of index values (integers) that define the order
in which input data is collected.
S.S.Culli Cull Index
Cull (remove) indexed elements from a list.
S.S.Cull Cull Pattern
Cull (remove) elements in a list using a repeating bit mask. The
bit mask is defined as a list of Boolean values. The bit mask is
repeated until all elements in the data list have been evaluated.
S.S.Dup Duplicate Data
Duplicate data a predefined number of times. Data can be
duplicated in two ways, either copies of the list are appended at
the end until the number of copies has been reached, or each
item is duplicated a number of times before moving on to the
next item.
S.S.Jitter Jitter
Randomly shuffles a list of values. The input list is reordered
based on random noise. Jittering is a good way to get a random
set with a good distribution. The jitter parameter sets radius of
the random noise. If jitter equals 0.5, then each item is allowed
to reposition itself randomly to within half the span of the entire
set.
S.S.Random Random
Generate a list of pseudo random numbers. The number
sequence is unique but stable for each seed value. If you do not
like a random distribution, try different seed values.
S.S.Range Range
Create a range of numbers. The numbers are spaced equally
inside a numeric domain. Use this component if you need to
create numbers between extremes. If you need control over the
interval between successive numbers, you should be using the
[Series] component.
S.S.Repeat Repeat Data
Repeat a pattern until it reaches a certain length.
S.S.Series Series
Create a series of numbers. The numbers are spaced
according to the {Step} value. If you need to distribute numbers
inside a fixed numeric range, consider using the [Range]
component instead.
S.T.Explode Explode Tree
Extract all the branches from a tree.
Sets
Tree
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S.T.Flatten Flatten Tree
Flatten a data tree by removing all branching information.
S.T.Flip Flip Matrix
Flip a matrix–like data tree by swapping rows and columns.
S.T.Graft Graft Tree
Typically, data items are stored in branches at specific index
values (0 for the first item, 1 for the second item, and so on and
so forth) and branches are stored in trees at specific branch
paths, for example: {0;1}, which indicates the second sub-branch
of the first main branch. Grafting creates a new branch for every
single data item.
S.T.Merge Merge
Merge a bunch of data streams.
S.T.Path Path Mapper
Perform lexical operations on data trees. Lexical operations are
logical mappings between data paths and indices which are
defined by textual (lexical) masks and patterns.
S.T.Prune Prune Tree
Removes all branches from a Tree that carry a special number of
Data items. You can supply both a lower and an upper limit for
branch pruning.
S.T.Simplify Simplify Tree
Simplify a tree by removing the overlap shared amongst all
branches.
S.T.TStat Tree Statistics
Get some statistics regarding a data tree.
S.T.Unflatten Unflatten Tree
Unflatten a data tree by moving items back into branches.
V.G.HexGrid Hexagonal
2D grid with hexagonal cells.
V.G.RecGrid Rectangular
2D grid with rectangular cells.
Vector
Grid
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V.G.SqGrid Square
2D grid with square cells
V.P.Pt Construct Point
Construct a point from {xyz} coordinates.
V.P.pDecon Deconstruct
Deconstruct a point into its component parts.
V.P.Dist Distance
Compute Euclidean distance between two point coordinates.
V.V.X Unit X
Unit vector parallel to the world {x} axis.
V.V.Y Unit Y
Unit vector parallel to the world {y} axis.
V.V.Vec2Pt Vector 2Pt
Create a vector between two points.
C.A.CP Control Points
Extract the nurbs control points and knots of a curve.
C.D.Divide Divide Curve
Divide a curve into equal length segments.
Point
Vector
Curve
Analysis
Division
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C.P.Cir Circle
Create a circle defined by base plane and radius.
C.P.Cir3Pt Circle 3Pt
Create a circle defined by three points.
C.P.CirCNR Circle CNR
Create a circle defined by center, normal and radius.
C.P.Line Line SDL
Create a line segment defined by start point, tangent and length.
C.P.Polygon Polygon
Create a polygon with optional round edges.
C.S.IntCrv Interpolate
Create an interpolated curve through a set of points.
C.S.KinkCrv Kinky Curve
Construct an interpolated curve through a set of points with a
kink angle threshold.
C.S.Nurbs Nurbs Curve
Construct a nurbs curve from control points.
C.S.PLine PolyLine
Create a polyline connecting a number of points.
C.U.Explode Explode
Explode a curve into smaller segments.
Primitive
Spline
Util
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<C.U.Join Join Curves
Join as many curves as possible.
C.U.Offset Offset
Offset a curve with a specified distance.
S.A.DeBrep Deconstruct Brep
Deconstruct a brep into its constituent parts.
S.F.Boundary Boundary Surfaces
Create planar surfaces from a collection of boundary edge
curves.
S.F.Extr Extrude
Extrude curves and surfaces along a vector.
S.F.ExtrPt Extrude Point
Extrude curves and surfaces to a point.
S.F.Loft Loft
Create a lofted surface through a set of section curves.
S.F.RevSrf Revolution
Create a surface of revolution.
S.F.Swp2 Sweep2
Create a sweep surface with two rail curves.
S.P.BBox Bounding Box
Solve oriented geometry bounding boxes.
Surface
Analysis
Freeform
Primitive
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S.U.SDivide Divide Surface
Generate a grid of {uv} points on a surface.
S.U.SubSrf Isotrim
Extract an isoparametric subset of a surface.
M.T.Voronoi Voronoi
Planar voronoi diagram for a collection of points.
T.A.RecMap Rectangle Mapping
Transform geometry from one rectangle into another.
T.A.ArrLinear Linear Array
Create a linear array of geometry.
T.M.Morph Box Morph
Morph an object into a twisted box.
Util
Mesh
Triangulation
Transform
Affine
Array
Morph
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235Index
T.M.SBox Surface Box
Create a twisted box on a surface patch.
D.C.HSL Colour HSL
Create a colour from floating point {HSL} channels.
D.D.Tag Text tags
A text tag component allows you to draw little Strings in the
viewport as feedback items. Text and location are specified as
input parameters. When text tags are baked they turn into Text
Dots.
D.D.Tag3D Text Tag 3D
Represents a list of 3D text tags in a Rhino viewport
D.P.Preview Custom Preview
Allows for customized geometry previews.
D.V.Points Point List
Displays details about lists of points.
Display
Color
Dimensions
Preview
Vector
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1.2.
1.2.5_the grasshopper definition.gh
1.3.
1.3.2.1_attractor definition.gh
1.3.3_operators and conditionals.gh
1.3.3.4_trigonometry components.gh
1.3.3.5_expressions.gh
1.3.4_domains and color.gh
1.3.5_booleans and logical operators.gh
1.4.
1.4.1.2_grasshopper spline components.gh
1.4.3_data matching.gh
1.4.4_list creation.gh
1.4.5_list visualization.gh
1.4.6_list management.gh
1.4.7_working with lists.gh
1.5.
1.5.1.3_morphing definition.gh
1.5.2.1_Data Tree Visualization.gh1.5.3_working with data trees.gh
1.5.3.6_weaving definition.gh
1.5.4_rail intersect definition.gh
1.6.
1.6.1_what is a mesh.gh
1.6.3_creating meshes.gh
1.6.6_working with meshes.gh
2.2. Grasshopper Example Files
These example files accompany the Grasshopper Primer, and are organized according to section.
The Grasshopper Primer V3.3
237Example Files
food4Rhino (WIP) is the new Plug-in Community Service by McNeel. As a user, find the newest Rhino Plug-ins,
Grasshopper Add-ons, Textures and Backgrounds, add your comments, discuss about new tools, get in contact
with the developers of these applications, share your scripts. http://www.food4rhino.com/
Grasshopper add-ons page http://www.grasshopper3d.com/page/addons-forgrasshopper 
DIVA-for-Rhino allows users to carry out a series of environmental performance evaluations of individual buildings
and urban landscapes. http://diva4rhino.com/
Element is a mesh geometry plugin for Grasshopper, enabling mesh creation, analysis, transformation,
subdivision, and smoothing. http://www.food4rhino.com/project/element
Firefly offers a set of comprehensive software tools dedicated to bridging the gap between Grasshopper and the
Arduino micro-controller. http://fireflyexperiments.com
GhPython is the Python interpreter component for Grasshopper that allows you to execute dynamic scripts of any
type. Unlike other scripting components, GhPython allows the use of rhinoscriptsyntax to start scripting without
needing to be a programmer. http://www.food4rhino.com/project/ghpython
2.3. Resources
There are many resources available to learn more about Grasshopper and parametric design concepts. There
are also over a hundred plugins and add-ons that extend Grasshopper’s functionality. Below are some of our
favorites.
Plug-in Communities
Add-ons We Love
The Grasshopper Primer V3.3
238Resources
http://www.food4rhino.com/
http://www.grasshopper3d.com/page/addons-forgrasshopper
http://diva4rhino.com/
http://www.food4rhino.com/project/element
http://fireflyexperiments.com
http://www.food4rhino.com/project/ghpython
HAL is a Grasshopper plugin for industrial robots programming supporting ABB, KUKA and Universal Robots
machines. http://hal.thibaultschwartz.com/
Extends Grasshopper’s ability to create and reference geometry including lights, blocks, and text objects. Also
enables access to information about the active Rhino document, pertaining to materials, layers, linetypes, and
other settings. http://www.food4rhino.com/project/human
Karamba is an interactive, parametric finite element program. It lets you analyze the response of 3-dimensional
beam and shell structures under arbitrary loads. http://www.karamba3d.com/
Kangaroo is a Live Physics engine for interactive simulation, optimization and form-finding directly within
Grasshopper. http://www.food4rhino.com/project/kangaroo
Fold panels using curved folding and control panel distribution on surfaces with a range of attractor systems.
http://www.food4rhino.com/project/robofoldkingkong
LunchBox is a plug-in for Grasshopper for exploring mathematical shapes, paneling, structures, and workflow.
http://www.food4rhino.com/project/lunchbox
Meshedit is a set of components which extend Grasshopper’s ability to work with meshes.
http://www.food4rhino.com/project/meshedittools
Parametric tools to create and manipulate rectangular grids, attractors and support creative morphing of parametric
patterns. http://www.food4rhino.com/project/pt-gh
The Grasshopper Primer V3.3
239Resources
http://hal.thibaultschwartz.com/
http://www.food4rhino.com/project/human
http://www.karamba3d.com/
http://www.food4rhino.com/project/kangaroo
http://www.food4rhino.com/project/robofoldkingkong
http://www.food4rhino.com/project/lunchbox
http://www.food4rhino.com/project/meshedittools
http://www.food4rhino.com/project/pt-gh
Platypus allows Grasshopper authors to stream geometry to the web in real time. It works like a chatroom for
parametric geometry, and allows for on-the-fly 3D model mashups in the web browser.
http://www.food4rhino.com/project/platypus
TT Toolbox features a range of different tools that we from the Core Studio at Thornton Tomasetti use on a regular
basis, and we thought some of you might appreciate these. http://www.food4rhino.com/project/tttoolbox
Weaverbird is a topological modeler that contains many of the known subdivision and transformation operators,
readily usable by designers. This plug-in reconstructs the shape, subdivides any mesh, even made by polylines,
and helps preparing for fabrication. http://www.giuliopiacentino.com/weaverbird/
The Firefly Primer This book is intended to teach the basics of electronics (using an Arduino) as well as various
digital/physical prototyping techniques to people new to the field. It is not a comprehensive book on electronics (as
there are already a number of great resources already dedicated to this topic). Instead, this book focuses on
expediting the prototyping process. Written by Andrew Payne. http://fireflyexperiments.com/resources/
Essential Mathematics Essential Mathematics uses Grasshopper to introduce design professionals to foundation
mathematical concepts that are necessary for effective development of computational methods for 3D modeling
and computer graphics. Written by Rajaa Issa.
http://www.rhino3d.com/download/rhino/5.0/EssentialMathematicsThirdEdition/
Generative Algorithms A series of books which is aimed to develop different concepts in the field of Generative
Algorithms and Parametric Design. Written by Zubin Khabazi. http://www.morphogenesism.com/media.html
Rhino Python Primer This primer is intended to teach programming to absolute beginners, people who have
tinkered with programming a bit or expert programmers looking for a quick introduction to the methods in Rhino.
Written by Skylar Tibbits. http://www.rhino3d.com/download/IronPython/5.0/RhinoPython101
Wolfram MathWorld is an online mathematics resource., assembled by Eric W. Weisstein with assistance from
thousands of contributors. Since its contents first appeared online in 1995, MathWorld has emerged as a nexus of
mathematical information in both the mathematics and educational communities. Its entries are extensively
referenced in journals and books spanning all educational levels. http://mathworld.wolfram.com/
Burry, Jane, and Mark Burry. The New Mathematics of Architecture. London: Thames & Hudson, 2010.
Burry, Mark. Scripting Cultures: Architectural Design and Programming. Chichester, UK: Wiley, 2011.
Additional Primers
General References
Further Reading
The Grasshopper Primer V3.3
240Resources
http://www.food4rhino.com/project/platypus
http://www.food4rhino.com/project/tttoolbox
http://www.giuliopiacentino.com/weaverbird/
http://fireflyexperiments.com/resources/
http://www.rhino3d.com/download/rhino/5.0/EssentialMathematicsThirdEdition/
http://www.morphogenesism.com/media.html
http://www.rhino3d.com/download/IronPython/5.0/RhinoPython101
http://mathworld.wolfram.com/
Hensel, Michael, Achim Menges, and Michael Weinstock. Emergent Technologies and Design: Towards a Biological
Paradigm for Architecture. Oxon: Routledge, 2010.
Jabi, Wassim. Parametric Design for Architecture. Laurence King, 2013.
Menges, Achim, and Sean Ahlquist. Computational Design Thinking. Chichester, UK: John Wiley & Sons, 2011.
Menges, Achim. Material Computation: Higher Integration in Morphogenetic Design. Hoboken, NJ: Wiley, 2012.
Peters, Brady, and Xavier De Kestelier. Computation Works: The Building of Algorithmic Thought. Wiley, 2013.
Peters, Brady. Inside Smartgeometry: Expanding the Architectural Possib ilities of Computational Design.
Chichester: Wiley, 2013.
Pottmann, Helmut, and Daril Bentley. Architectural Geometry. Exton, PA: Bentley Institute, 2007.
Sakamoto, Tomoko, and Albert Ferré. From Control to Design: Parametric/algorithmic Architecture. Barcelona: Actar-
D, 2008.
Woodbury, Robert. Elements of Parametric Design. London: Routledge, 2010.
The Grasshopper Primer V3.3
241ResourcesThe Grasshopper Primer V3.3
242About This Primer
	About
	Grasshopper - an Overview
	Grasshopper in Action
	Foundations
	Hello Grasshopper!
	Installing and Launching Grasshopper
	The Grasshopper UI
	Talking to Rhino
	Anatomy of a Grasshopper Definition
	Grasshopper Object Types
	Grasshopper Component Parts
	Data Types
	Wiring Components
	The Grasshopper Definition
	Building Blocks of Algorithms
	Points Planes & Vectors
	Working With Attractors
	Mathematics, Expressions & Conditionals
	Domains & Color
	Booleans and Logical Operators
	Designing with Lists
	Curve Geometry
	What is a List?
	Data Stream Matching
	Creating Lists
	List Visualization
	List Management
	Working with Lists
	Designing with Data Trees
	Surface Geometry
	What is a Data Tree?
	Creating Data Trees
	Working with Data Trees
	Getting Started with Meshes
	What is a Mesh?
	Understanding Topology
	Creating Meshes
	Mesh Operations
	Mesh Interactions
	Working with Mesh Geometry
	Extensions
	Element*
	Introduction
	Half Edge Data
	Components
	Exercise
	Architectural Case Study
	Appendix
	Index
	Example Files
	Resources
	About This Primer